Content-Type: multipart/mixed; boundary="-------------0504130757432" This is a multi-part message in MIME format. ---------------0504130757432 Content-Type: text/plain; name="05-134.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-134.comments" 52 pages, 4 figures, Latex2e ---------------0504130757432 Content-Type: text/plain; name="05-134.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-134.keywords" Exclusion process, stationary TASEP, Random matrices ---------------0504130757432 Content-Type: application/postscript; name="FinalTasep.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="FinalTasep.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.94a Copyright 2003 Radical Eye Software %%Title: FinalTasep.dvi %%CreationDate: Wed Apr 13 14:26:42 2005 %%Pages: 52 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMBX10 CMR10 CMMI10 CMMI8 CMSY10 CMR8 %%+ CMBX12 CMTI12 CMMI12 CMEX10 CMSY8 CMR6 CMSY6 CMMI6 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 -t A4 -z FinalTasep %DVIPSParameters: dpi=8000 %DVIPSSource: TeX output 2005.04.13:1426 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro 0 0 %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 90 /Z put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /mu put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 61 /slash put dup 78 /N put dup 99 /c put dup 107 /k put dup 114 /r put dup 119 /w put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 54 /six put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 3645B82392D5CAE11A7CB49D7E2E82DCD485CBA17D1AFFF95F4224CF7ECEE45C BFB7C8C77C22A01C345078D28D3ECBF804CDC2FE5025FA0D05CCC5EFC0C4F87E CBED13DDDF8F34E404F471C6DD2E43331D73E89BBC71E7BF889F6293793FEF5A C9DD3792F032E37A364C70914843F7AA314413D022AE3238730B420A7E9D0CF5 D0E24F501451F9CDECE10AF7E14FF15C4F12F3FCA47DD9CD3C7AEA8D1551017D 23131C09ED104C052054520268A4FA3C6338BA6CF14C3DE3BAF2EA35296EE3D8 D6496277E11DFF6076FE64C8A8C3419FA774473D63223FFA41CBAE609C3D976B 93DFB4079ADC7C4EF07303F93808DDA9F651F61BCCF79555059A44CBAF84A711 6D98083CEF58230D54AD486C74C4A257FC703ACF918219D0A597A5F680B606E4 EF94ADF8BF91A5096A806DB64EC96636A98397D22A74932EB7346A9C4B5EE953 CB3C80AA634BFC28AA938C704BDA8DC4D13551CCFE2B2784BE8BF54502EBA9AF D49B79237B9C56310550BC30E9108BB06EAC755D6AA4E688EFE2A0AAB17F20FE 00CD0BFF1B9CB6BDA0FA3A29A3117388B6686657A150CE6421FD5D420F4F7FB5 B0DAA1BA19D638676E9CF159AC7325EF17B9F74E082BEF75E10A31C7011C0FFA 99B797CE549B5C45238DD0FADD6B99D233AC69282DF0D91EA2DBD08CE0083904 A6D968D5AE3BD159D01BDFF42D16111BC0A517C66B43972080D9DD4F3B9AE7FB 11B035CE715C1218B2D779761D8D7E9DEBE277531BD58F313EBD27E33BEF9DC5 50C7821A8BBC3B9FDF899D7EAA0B94493B97AFEAC503EB5ED7A7AB6CCCC950AF F6A29F479A2D64381EEF89EDD5FF9B5592CE8F4EE8E3BA1E6E9AA35F5FC82629 E25D1F36DF188864FB57F960F508173D284660BD33334C022E06D6B3CDBC272B C8FD6324218C00DD1EF4C0BB2098AE7AC732209CA00AFBDDCA81147BF7673CB0 59C3EFC6DCE1B7F5FAB8C1610BF6CB38917B9D477E87726C65A520B2BE0595A9 5F960B111E818479CE2811F45966785F205D344E4DFE632C06705CBB5B4EEC20 E731D7DA971EABC6923DED1B3A663BB16F453CA20BA20D3A281DD2BB0ADAA592 F1B999078F84D0D1A4DD09AA5DF4302706CEDECCCFB5848F36129332C1E9A869 54116686708EC8A88F7781E0C1A46B8B574D361E4074793BBF153765FA9C0D7F 84E95E19CCFFC94EF6B1B81410768FB00B807C36EB7B60E8B6ED08482E48A374 ACC5DD9FA74812ED51E4F3A24242557F0DD3531BA0B8D7BD21B6AFDC54EEFEC6 FF6C3753FFD7A707F66FFAB2D8F59A6C04A297B207D3E8EDE2ABB5B330571A27 27C0D2BF072FF52A98A14AF6E28E83C74E03C10F3B81A3F277B8B06F9647E2C8 748F5423DF06DFBD8A517659A7643478C383C67187BA84CAB4D8A3126AE67DCF 0CFCC199CB90813C574E89E1C0F36910617BF5A6B7346518B4A8AD5413786272 C4150BA540BBFA790937A21E69E30E660F47954569BCB2403BECA5EF2F0FBDA7 A30F92EE880A46FA758722A0857F6BE3FC83451BFA922B5FE7BB3BD45F6742CA 1F49D028C96BDF89DAEECBF33BF189E8C5509AF397FB3BCD64972870FD3F3C69 00D36D213E8BCCF605B9515F632B4838A823273C6D2D93789BBE6524E0CD4551 6D1BFCA0BC5EDF2D025B7CA2A6EC2BB85B2427947ED75442F121CD9A3BB94F76 7A5C335020C575AA8E73CBA2010078A140F542EA5F8AEBDB01BFEB3D7DF9E8D3 DA6EC07AEC0C5D3D603839EB10E4A666D010C80DD4FBD0857FE2E643216B8333 2FA960AA75783C511880043B40C12194F6C56D5EC666269CD720FB2E9746EE39 770C31C2D6D6E6445F3ACBFB3E46742B637EFEA252BDA1436D3B4F5198B5587F 758EC8390CF47722E4E56C8FE755EE7FC2796BC09B474B2573994941AFC33BB5 31A22EA7E243745B2B427AF2188349A0F4D04E49BE1F1F45A004727025345055 CE7C5D93D1E8D0CBAEA1C9191BD7FC8BE718D64D42788E1C77E2A2B478ACF0AB 1D09EDE6E1D0E95BD8988065EA9017895978CE9A7A95A8A47FFEDEB4B93630A9 78DE04CF9237781482E631D4189A4F43E4569FB1D708F16A88EEC0EC6F38090A 8263E6A66FDD594EB03A26C95FEE356D32808E079AFA1DDE4EFE912D12DEB7BB B6B02EAE2E73756770B8093DCDB2720355930A90B9FFE95B02B3D698D1E2B752 4FFC6374AD30F890EE82B3A6FAA4F7C8F13212D228E59201E5E39CAADD2603BF 9878D92BD17470441D10D7810186046EE1DBE3048F8DE40195DBAA40EDFFA5A0 8EB47CDA7D4FFD19CB8792718BC5D22B64864453CF61EE573887BCC4D8789226 28A8D3648F411905E45A5828FE606E6904B70062AE5269CBF9CEB514A37F43CA 3D9E56E6BCE9BF145AFA111ED8BCA17641FEF0550849FB6D26D294193BE44296 D3B8E185E71FF0C8B3C7D589BE308B9B8ABAD66DDB3DF42AE65B3A532DFACA80 F704B726DF89944F7C8A4D5D62F3B8C62852D0CC4E67E4C7826CDBB0952E3B83 B5D9645974B597E54E 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 70 /F put dup 79 /O put dup 91 /union put dup 98 /floorleft put dup 99 /floorright put dup 106 /bar put dup 112 /radical put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 102 /tildewider put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 525003F3DBE5BF07B2E83E66B7F97DDD7CE0EEB75A78BD9227BF359D002B6ADB 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 69 /E put dup 71 /G put dup 72 /H put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 91 /bracketleft put dup 97 /a put dup 99 /c put dup 100 /d put dup 105 /i put dup 109 /m put dup 115 /s put dup 120 /x put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 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Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 20 /kappa put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 59 /comma put dup 61 /slash put dup 64 /partialdiff put dup 76 /L put dup 77 /M put dup 78 /N put dup 82 /R put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F00F963068B8B731A88D7740B0DDAED1B3F82 7DB9DFB4372D3935C286E39EE7AC9FB6A9B5CE4D2FAE1BC0E55AE02BFC464378 77B9F65C23E3BAB41EFAE344DDC9AB1B3CCBC0618290D83DC756F9D5BEFECB18 2DB0E39997F264D408BD076F65A50E7E94C9C88D849AB2E92005CFA316ACCD91 FF524AAD7262B10351C50EBAD08FB4CD55D2E369F6E836C82C591606E1E5C73F DE3FA3CAD272C67C6CBF43B66FE4B8677DAFEEA19288428D07FEB1F4001BAA68 7AAD6DDBE432714E799CFA49D8A1A128F32E8B280524BC8041F1E64ECE4053C4 9F0AEC699A75B827002E9F95826DB3F643338F858011008E338A899020962176 CF66A62E3AEF046D91C88C87DEB03CE6CCDF4FB651990F0E86D17409F121773D 6877DF0085DFB269A3C07AA6660419BD0F0EF3C53DA2318BA1860AB34E28BAC6 E82DDB1C43E5203AC9DF9277098F2E42C0F7BD03C6D90B629DE97730245B8E8E 8903B9225098079C55A37E4E59AE2A9E36B6349FA2C09BB1F5F4433E4EEFC75E 3F9830EB085E7E6FBE2666AC5A398C2DF228062ACF9FCA5656390A15837C4A99 EC3740D873CFEF2E248B44CA134693A782594DD0692B4DBF1F16C4CDECA692C4 0E44FDBEF704101118BC53575BF22731E7F7717934AD715AC33B5D3679B784C9 4046E6CD3C0AD80ED1F65626B14E33CFDA6EB2825DC444FA6209608D3976637A 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3B054860006DEA20685F5E2937C95B50D07C6316ABC08495EF319B36F8E48FD0 DA482F82D2D981C70ADA2E467608364EC664D151BCE2FDF571BA63FC926CCD72 052D4F83933A9585C0E033CE579227FB5BE2D58E048044B5BB6869ADFC2E03C7 AC062D709CC07D8E4377D75506A96CEC17A90E1FBEF6E3778DE10A910F346C4E D33D753F6EC32D638BBDACC51CF22379D24C2A0780333AB3CB5A612C14381B3F B7E42E841B123CBBBA6C33C080F9C6D370A91F29A4CA724B9809042BE9B2EDEA 366C4D17F80C7923C9DFEB1C184DA0C0396BB3AE8446D1FDD7C76397076DDEAF 4FB188394902EAC1B9126016723FBAAD27AD91C44C28F1D579989D8C3B5D6AD1 1647504F2F6C38F1786DB3C9DE2BD6CA02D74451FDF6161DCD51706D62B383E1 A1E9F0D97F9579FFDC844710C4D9EEBC5F048B0811B8F3BF2BBA45C9A57E7B27 48BD0F25113B884DA2D1596E14FCC7FE2B528806F4D61BB6A6D34E85AFF5193F 7B396AE95ECAD67587C51633CB0F12327BB56E36C5E7EFF9437D80C2B337A480 3EAD6F51E3475A8638B233545B633EABEA2EA89D7CD3A073FD32A1C5DFEE4ED7 402DCC2E2CC4AC399619B272A5456E735F1044685EF0EC8BA6C242D5A029204C 6E32D78226FC39E9222F7AD77F162827808F98CEB740B798BAAA7BB865A33D74 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cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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Fw(\))371 b Fm(\024)e Fn(t)p Fm(g)14593 40712 y Fj(\001)15570 41788 y Fw(=)16951 39516 y Fj(\()18021 40878 y Fl(P)19034 41077 y Fq(T)-78 b(A)20393 39802 y Fj(\000)21002 40878 y Fm(f)p Fn(\034)444 b Fm(\000)295 b Fw(1)369 b Fm(\024)h Fn(h)27174 41077 y Fs(t)27568 40878 y Fw(\()p Fn(j)75 b Fw(\))p Fm(g)29857 39802 y Fj(\001)30466 40878 y Fn(;)1301 b Fw(if)434 b(\()p Fm(\000)p Fw(1\))36015 40396 y Fs(\034)102 b Fq(+)p Fs(j)38123 40878 y Fw(=)368 b Fm(\000)p Fw(1)p Fn(;)18021 42804 y Fl(P)19034 43003 y Fq(T)-78 b(A)20393 41728 y Fj(\000)21002 42804 y Fm(f)p Fn(\034)444 b Fm(\000)295 b Fw(2)369 b Fm(\024)h Fn(h)27174 43003 y Fs(t)27568 42804 y Fw(\()p Fn(j)75 b Fw(\))p Fm(g)29857 41728 y Fj(\001)30466 42804 y Fn(;)1301 b Fw(if)434 b(\()p Fm(\000)p Fw(1\))36015 42322 y Fs(\034)102 b Fq(+)p Fs(j)38123 42804 y Fw(=)368 b(1)p Fn(:)43929 41788 y Fw(\(2.6\))5396 45726 y(~)5383 46077 y Fn(h)o Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))399 b(is)e(not)g(suc)-36 b(h)397 b(a)g(con)-36 b(v)g(enien)g(t)397 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Fw(\()p Fn(m;)221 b(n)p Fw(\))p Fm(j)369 b(\025)g Fn(u)p Fm(gj)p Fn(\020)25853 55382 y Fj(\001)26832 56458 y Fm(\024)g Fn(\020)28802 56657 y Fq(+)29589 56458 y Fn(e)30192 55910 y Fi(\000)p Fs(u=)p Fq(\(1)p Fi(\000)p Fs(\032)p Fq(\))34706 56458 y Fw(+)295 b Fn(\020)36581 56657 y Fi(\000)37368 56458 y Fn(e)37971 55910 y Fi(\000)p Fs(u=\032)40255 56458 y Fn(:)3313 b Fw(\(2.7\))3432 59349 y Fo(Pr)-66 b(o)g(of.)649 b Fw(W)-108 b(e)347 b(\014x)g(the)g(endp)36 b(oin)-36 b(t)346 b(\()p Fn(m;)221 b(n)p Fw(\).)550 b(Let)347 b Fn(T)26506 58867 y Fs(\020)27036 59349 y Fw(\()p Fn(!)48 b Fw(\))346 b(b)36 b(e)348 b(the)e(passage)i(time)g(for)g Fn(w)43423 59548 y Fs(\020)43952 59349 y Fw(\()p Fn(i;)221 b(j)75 b Fw(\))3432 60954 y(and)469 b(let)h Fn(!)8770 60472 y Fs(\020)8722 61282 y Fq(max)10999 60954 y Fw(b)36 b(e)470 b(a)g(maximizing)i(path)d(from)i(\(0)p Fn(;)221 b Fw(0\))471 b(to)f(\()p Fn(m;)221 b(n)p Fw(\).)688 b(Then)470 b Fn(G)41122 60472 y Fs(\020)41652 60954 y Fw(\()p Fn(m;)221 b(n)p 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Fw(\))17751 67387 y Fm(\024)1107 b Fn(T)20834 66838 y Fq(0)21359 67387 y Fw(\()p Fn(!)22723 66838 y Fq(0)22675 67715 y(max)24482 67387 y Fw(\))295 b Fm(\000)g Fn(T)27554 66838 y Fs(\020)28084 67387 y Fw(\()p Fn(!)29448 66838 y Fq(0)29400 67715 y(max)31206 67387 y Fw(\))17751 70663 y Fm(\024)20305 68910 y Fs(\020)20719 69029 y Fg(+)19891 69401 y Fj(X)20089 72201 y Fs(i)p Fq(=1)22031 70663 y Fn(w)36 b Fw(\()p Fn(i;)221 b Fw(0\))295 b(+)27692 68910 y Fs(\020)28106 69029 y Ff(\000)27287 69401 y Fj(X)27429 72201 y Fs(j)51 b Fq(=1)29427 70663 y Fn(w)36 b Fw(\(0)p Fn(;)221 b(j)75 b Fw(\))p Fn(;)10318 b Fw(\(2.8\))24692 75163 y(8)p eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 3432 7638 a Fw(where)320 b(in)g(the)f(\014rst)g(inequalit)-36 b(y)322 b(w)-36 b(e)320 b(used)f(that)h Fn(!)27314 7156 y Fs(\020)27266 7966 y Fq(max)29393 7638 y Fw(is)g(a)g(maximizer)i(of)f Fn(T)40078 7156 y Fs(\020)40607 7638 y Fw(.)541 b(Similarly)4900 10571 y Fn(G)5926 10023 y Fq(0)6451 10571 y Fw(\()p 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26146 45 878 v 45708 25313 781 45 v 45708 26146 V 46487 26146 45 878 v 3432 28858 a Fh(De\014nition)810 b(2.3.)692 b Fn(h)14482 28376 y Fq(0)15007 28858 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))714 b Fo(is)g(the)e(height)h(function)e(as)j(given)e(in)g (\(2.5\),)775 b(wher)-66 b(e)3432 30463 y Fn(G)p Fw(\()p Fn(m;)221 b(n)p Fw(\))345 b Fo(is)g(r)-66 b(eplac)g(e)g(d)344 b(by)h Fn(G)16998 29981 y Fq(0)17523 30463 y Fw(\()p Fn(m;)221 b(n)p Fw(\))346 b Fo(with)f(the)g(c)-66 b(orr)g(esp)g(onding) 342 b(p)-66 b(assage)346 b(times)e Fn(w)43028 30662 y Fq(0)43554 30463 y Fw(\()p Fn(i;)221 b(j)75 b Fw(\))p Fo(.)5383 33175 y Fw(It)608 b(follo)-36 b(ws)611 b(from)e(the)f(iden) -36 b(tit)g(y)608 b(\(2.6\))h(and)f(Prop)36 b(osition)609 b(2.2)g(that)f(for)i Fn(")666 b(>)g Fw(0,)3432 34781 y(indep)36 b(enden)-36 b(t)431 b(of)j Fn(t)p Fw(,)g(one)f(can)h(c)-36 b(ho)36 b(ose)434 b Fn(d)p Fw(\()p Fn(")p Fw(\))f(suc)-36 b(h)432 b(that)3921 37714 y(\(1)295 b Fm(\000)g Fn(")p Fw(\))p Fl(Q)8948 36638 y Fj(\000)9561 37714 y Fm(f)p Fn(h)10974 37166 y Fq(0)11500 37714 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))370 b Fm(\024)f Fn(t)295 b Fm(\000)g Fn(d)p Fw(\()p Fn(")p Fw(\))p Fm(g)21173 36638 y Fj(\001)22151 37714 y Fm(\024)369 b Fn(F)24395 37913 y Fs(w)25149 37714 y Fw(\()p Fn(s;)221 b(t)p Fw(\))369 b Fm(\024)g Fl(Q)30730 36638 y Fj(\000)31343 37714 y Fm(f)p Fn(h)32756 37166 y Fq(0)33282 37714 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))370 b Fm(\024)f Fn(t)295 b Fw(+)g Fn(d)p Fw(\()p Fn(")p Fw(\))p Fm(g)42934 36638 y Fj(\001)43837 37714 y Fw(+)g Fn(";)43279 39319 y Fw(\(2.11\))3432 40924 y(uniformly)434 b(in)f Fn(j)r(;)221 b(\034)148 b Fw(.)580 b(Therefore,)435 b(setting)6629 43858 y Fn(F)7652 43309 y Fq(0)7471 44186 y Fs(w)8225 43858 y Fw(\()p Fn(s;)221 b(t)p Fw(\))1107 b(=)f Fl(Q)15260 42782 y Fj(\000)15874 43858 y Fm(f)p Fn(h)17287 43309 y Fq(0)17812 43858 y Fw(\()p Fm(b)p Fw(\(1)296 b Fm(\000)f Fw(2)p Fn(\032)p Fw(\))p Fn(t)g Fw(+)g(2)p Fn(w)36 b(\037)28018 43309 y Fq(1)p Fs(=)p Fq(3)29484 43858 y Fn(t)29954 43309 y Fq(2)p Fs(=)p Fq(3)31421 43858 y Fm(c)p Fn(;)10907 b Fw(\(2.12\))14127 45978 y Fm(b)p Fw(\(1)296 b Fm(\000)f Fw(2)p Fn(\037)p Fw(\))p Fn(t)g Fw(+)g(2)p Fn(w)36 b Fw(\(1)295 b Fm(\000)g Fw(2)p Fn(\032)p Fw(\))p Fn(\037)28577 45430 y Fq(1)p Fs(=)p Fq(3)30044 45978 y Fn(t)30514 45430 y Fq(2)p Fs(=)p Fq(3)32276 45978 y Fm(\000)g Fn(s\037)35032 45430 y Fq(2)p Fs(=)p Fq(3)36498 45978 y Fn(t)36968 45430 y Fq(1)p Fs(=)p Fq(3)38435 45978 y Fm(c)p Fw(\))369 b Fm(\024)g Fn(t)p Fm(g)42436 44902 y Fj(\001)43045 45978 y Fn(;)3432 48911 y Fw(one)433 b(has)7827 51845 y(\(1)295 b Fm(\000)g Fn(")p Fw(\))p Fn(F)12744 51296 y Fq(0)12563 52173 y Fs(w)13317 50769 y Fj(\000)13926 51845 y Fn(s;)221 b(t)295 b Fm(\000)h Fn(d)p Fw(\()p Fn(")p Fw(\))19512 50769 y Fj(\001)20489 51845 y Fm(\024)369 b Fn(F)22733 52044 y Fs(w)23487 51845 y Fw(\()p Fn(s;)221 b(t)p Fw(\))369 b Fm(\024)g Fn(F)28958 51296 y Fq(0)28777 52173 y Fs(w)29531 50769 y Fj(\000)30140 51845 y Fn(s;)221 b(t)295 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def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 170 moveto 0 0 lineto 632 0 lineto 632 170 lineto closepath clip newpath -35.3 441.0 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proc char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor 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b(y)410 b(\()14175 56771 y Fq(1)p 14175 56988 471 54 v 14175 57752 a(2)15023 57294 y Fw(+)246 b Fn(ik)45 b Fw(\))17952 56812 y Fi(\000)p Fq(1)19619 57294 y Fw(and)408 b Fn(T)22885 57493 y Fi(\000)24082 57294 y Fw(b)-36 b(y)410 b(\()26504 56771 y Fq(1)p 26504 56988 V 26504 57752 a(2)27352 57294 y Fm(\000)246 b Fn(ik)45 b Fw(\))30302 56812 y Fi(\000)p Fq(1)31560 57294 y Fw(.)570 b(Eigenfunctions)409 b(of)h Fn(T)43490 57493 y Fq(+)44278 57294 y Fw(,)415 b Fn(T)45816 57493 y Fi(\000)3432 58899 y Fw(are)433 b(the)g(exp)36 b(onen)-36 b(tials)434 b Fn( )16171 59098 y Fs(a)16726 58899 y Fw(\()p Fn(x)p Fw(\))369 b(=)f Fn(e)20829 58417 y Fi(\000)p Fs(ax)22646 58899 y Fw(.)578 b(F)-108 b(or)433 b Fn(a)369 b(<)28481 58376 y Fq(1)p 28481 58593 V 28481 59357 a(2)29517 58899 y Fw(one)434 b(has)18562 62551 y Fn(T)19324 62750 y Fq(+)20112 62551 y Fn( )20958 62750 y Fs(a)21512 62551 y Fw(\()p Fn(x)p Fw(\))369 b(=)26342 61652 y(1)p 25146 62245 3043 54 v 25279 62975 a Fq(1)p 25279 63193 471 54 v 25279 63956 a(2)26177 63498 y Fm(\000)295 b Fn(a)28321 62551 y( )29167 62750 y Fs(a)29722 62551 y Fw(\()p Fn(x)p Fw(\))11806 b(\(3.11\))3432 66489 y(and)433 b(for)h Fn(a)369 b(>)f Fm(\000)11546 65966 y Fq(1)p 11546 66184 V 11546 66947 a(2)12583 66489 y Fw(one)434 b(has)18392 70141 y Fn(T)19154 70340 y Fi(\000)19942 70141 y Fn( )20788 70340 y Fs(a)21343 70141 y Fw(\()p Fn(x)p Fw(\))368 b(=)26161 69242 y(1)p 24976 69835 3021 54 v 25109 70565 a Fq(1)p 25109 70783 471 54 v 25109 71547 a(2)26007 71089 y Fw(+)295 b Fn(a)28129 70141 y( )28975 70340 y Fs(a)29530 70141 y Fw(\()p Fn(x)p Fw(\))p Fn(:)11637 b Fw(\(3.12\))24367 75163 y(13)p eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 3432 7638 a Fw(F)-108 b(or)433 b Fm(j)p Fn(d)p Fm(j)369 b(\024)g Fn(m)433 b Fw(let,)h(as)g(an)f(op)36 b(erator)434 b(in)g Fn(L)23803 7156 y Fq(2)24328 7638 y Fw(\()p Fl(R)p Fw(\),)21017 10185 y Fn(K)22124 10384 y Fs(m;d)24126 10185 y Fw(=)368 b Fn(LP)27229 10384 y Fi(\000)28017 10185 y Fn(R)14272 b Fw(\(3.13\))3432 12732 y(with)13646 14337 y Fn(L)369 b Fw(=)f Fn(T)17223 13788 y Fs(m)p Fq(+)p Fs(d)17042 14665 y Fq(+)19327 14337 y Fn(T)20270 13658 y Fi(\000)p Fq(\()p Fs(m)p Fi(\000)p Fs(d)p Fq(\))20089 14627 y Fi(\000)23837 14337 y Fn(;)1522 b(R)380 b Fw(=)369 b Fn(T)29414 13658 y Fi(\000)p Fq(\()p Fs(m)p Fq(+)p Fs(d)p Fq(\))29233 14627 y(+)32981 14337 y Fn(T)33924 13788 y Fs(m)p Fi(\000)p Fs(d)33743 14665 y Fi(\000)36028 14337 y Fn(:)6890 b Fw(\(3.14\))3432 16496 y(It)433 b(follo)-36 b(ws)436 b(that,)d(for)h Fn(a)369 b Fm(2)g Fw(\()p Fm(\000)p Fw(1)p Fn(=)p Fw(2)p Fn(;)221 b Fw(1)p Fn(=)p Fw(2\),)16884 19042 y(\()p Fn(R)11 b( )19237 19241 y Fi(\000)p Fs(a)20523 19042 y Fw(\)\()p Fn(x)p Fw(\))1106 b(=)h Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))p Fn( )29531 19241 y Fi(\000)p Fs(a)30818 19042 y Fw(\()p Fn(x)p Fw(\))p Fn(;)10349 b Fw(\(3.15\))17206 20980 y(\()p Fn(L)18597 20431 y Fi(\003)19122 20980 y Fn( )19968 21179 y Fs(a)20523 20980 y Fw(\)\()p Fn(x)p Fw(\))1106 b(=)h Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))28685 20431 y Fi(\000)p Fq(1)29943 20980 y Fn( )30789 21179 y Fs(a)31343 20980 y Fw(\()p Fn(x)p Fw(\))3432 23526 y(where)19410 25710 y Fn(Z)g Fw(\()p Fn(a)p Fw(\))369 b(=)23983 24734 y(\()24622 24211 y Fq(1)p 24622 24429 471 54 v 24622 25192 a(2)25521 24734 y Fw(+)294 b Fn(a)p Fw(\))28016 24252 y Fs(m)p Fq(+)p Fs(d)p 23973 25405 6158 54 v 23973 26658 a Fw(\()24612 26135 y Fq(1)p 24612 26352 471 54 v 24612 27116 a(2)25510 26658 y Fm(\000)h Fn(a)p Fw(\))28027 26274 y Fs(m)p Fi(\000)p Fs(d)30263 25710 y Fn(:)12655 b Fw(\(3.16\))5383 28703 y(F)-108 b(or)304 b(later)h(use)f(w)-36 b(e)305 b(pro)-36 b(vide)305 b(a)g(represen)-36 b(tation)303 b(of)j(the)e(k)-36 b(ernel)304 b(of)i Fn(K)37837 28902 y Fs(m;d)39470 28703 y Fw(.)535 b(This)305 b(k)-36 b(ernel)3432 30308 y(has)480 b(a)g(singular)h(part,) 492 b(whic)-36 b(h)480 b(is)g(concen)-36 b(trated)479 b(on)h(the)g(diagonal)h Fm(f)p Fn(x)449 b Fw(=)f Fn(y)48 b Fm(g)p Fw(.)719 b(In)480 b(the)3432 31913 y(computations)474 b(only)i(the)f(regular)g(part)g(will)h(b)36 b(e)475 b(used,)486 b(hence)474 b(only)i(it)f(is)h(displa)-36 b(y)g(ed.)3432 33518 y(Since)632 b(in)h(F)-108 b(ourier)633 b(space)g Fn(T)17879 33717 y Fq(+)18666 33518 y Fw(,)684 b(resp.)633 b Fn(T)23787 33717 y Fi(\000)24574 33518 y Fw(,)683 b(is)634 b(the)e(op)36 b(erator)634 b(of)g(m)-36 b(ultiplication)633 b(b)-36 b(y)3432 35123 y(\()4071 34600 y Fq(1)p 4071 34818 V 4071 35581 a(2)5003 35123 y Fw(+)328 b Fn(ik)45 b Fw(\))8014 34641 y Fi(\000)p Fq(1)9272 35123 y Fw(,)496 b(resp.)483 b(\()13932 34600 y Fq(1)p 13932 34818 V 13932 35581 a(2)14864 35123 y Fm(\000)330 b Fn(ik)45 b Fw(\))17898 34641 y Fi(\000)p Fq(1)19155 35123 y Fw(,)496 b(with)484 b(the)e(c)-36 b(hange)484 b(of)g(v)-72 b(ariable)36255 34600 y Fq(1)p 36255 34818 V 36255 35581 a(2)37188 35123 y Fm(\000)329 b Fn(ik)499 b Fw(=)453 b Fn(z)389 b Fw(+)328 b Fn(\032)484 b Fw(w)-36 b(e)3432 36728 y(obtain)496 b(an)h(in)-36 b(tegral)497 b(expression)g(for)g(the)f(regular)h(part)f (of)h Fn(L)g Fw(and)f Fn(R)11 b Fw(.)768 b(Let)496 b(\000)43094 36927 y Fs(p)44119 36728 y Fw(b)36 b(e)497 b(a)3432 38333 y(path)399 b(around)h(the)g(p)36 b(ole)401 b Fn(p)f Fw(orien)-36 b(ted)400 b(an)-36 b(ti-clo)36 b(c)-36 b(kwise.)569 b(Then)400 b(the)g(regular)h(part)f(of)i(the)3432 39939 y(k)-36 b(ernels)433 b(are)10375 42746 y Fn(L)p Fw(\()p Fn(x;)221 b(y)48 b Fw(\))369 b(=)17288 41847 y(1)p 16157 42440 2913 54 v 16157 43657 a Fm(\000)p Fw(2)p Fn(\031)48 b(i)19203 42746 y(e)19806 42197 y Fq(\(1)p Fs(=)p Fq(2)p Fi(\000)p Fs(\032)p Fq(\)\()p Fs(x)p Fi(\000)p Fs(y)32 b Fq(\))25708 42746 y Fn(I)26282 42945 y Fs(m;d)27915 42746 y Fw(\()p Fn(x)295 b Fm(\000)g Fn(y)48 b Fw(\))p Fn(;)1522 b(x)296 b Fm(\000)f Fn(y)417 b(>)368 b Fw(0)p Fn(;)3620 b Fw(\(3.17\))3432 45742 y(where)12090 47849 y Fn(I)12664 48048 y Fs(m;d)14297 47849 y Fw(\()p Fn(x)295 b Fm(\000)g Fn(y)48 b Fw(\))369 b(=)20102 46041 y Fj(Z)20840 49048 y Fq(\000)21428 49171 y Fg(1)p Ff(\000)p Fe(\032)23230 47849 y Fw(d)p Fn(z)59 b(e)25219 47300 y Fi(\000)p Fs(z)37 b Fq(\()p Fs(x)p Fi(\000)p Fs(y)32 b Fq(\))30252 46950 y Fw(\()p Fn(z)354 b Fw(+)295 b Fn(\032)p Fw(\))34200 46468 y Fs(m)p Fi(\000)p Fs(d)p 29104 47543 8347 54 v 29104 48760 a Fw(\(1)g Fm(\000)h Fn(\032)f Fm(\000)g Fn(z)59 b Fw(\))35347 48376 y Fs(m)p Fq(+)p Fs(d)37584 47849 y Fn(;)5334 b Fw(\(3.18\))3432 50887 y(and)433 b(similarly)10833 53952 y Fn(R)11 b Fw(\()p Fn(x;)221 b(y)48 b Fw(\))369 b(=)17346 53054 y(1)p 16732 53647 1880 54 v 16732 54864 a(2)p Fn(\031)48 b(i)18744 53952 y(e)19347 53404 y Fq(\(1)p Fs(=)p Fq(2)p Fi(\000)p Fs(\032)p Fq(\)\()p Fs(x)p Fi(\000)p Fs(y)32 b Fq(\))25408 53617 y Fw(~)25249 53952 y Fn(I)25823 54151 y Fs(m;d)27456 53952 y Fw(\()p Fn(y)343 b Fm(\000)295 b Fn(x)p Fw(\))p Fn(;)1523 b(x)295 b Fm(\000)g Fn(y)417 b(<)369 b Fw(0)p Fn(;)4078 b Fw(\(3.19\))3432 56839 y(where)12817 58609 y(~)12659 58945 y Fn(I)13233 59144 y Fs(m;d)14865 58945 y Fw(\()p Fn(y)343 b Fm(\000)296 b Fn(x)p Fw(\))368 b(=)20671 57137 y Fj(Z)21409 60144 y Fq(\000)21997 60263 y Ff(\000)p Fe(\032)23393 58945 y Fw(d)p Fn(z)59 b(e)25382 58396 y Fs(z)37 b Fq(\()p Fs(y)32 b Fi(\000)p Fs(x)p Fq(\))28535 58046 y Fw(\(1)295 b Fm(\000)h Fn(\032)f Fm(\000)g Fn(z)59 b Fw(\))34778 57564 y Fs(m)p Fq(+)p Fs(d)p 28535 58639 8347 54 v 29683 59856 a Fw(\()p Fn(\032)295 b Fw(+)f Fn(z)59 b Fw(\))33630 59472 y Fs(m)p Fi(\000)p Fs(d)37015 58945 y Fn(:)5903 b Fw(\(3.20\))3432 62053 y(Let)433 b Fn(P)6600 62252 y Fs(u)7634 62053 y Fw(b)36 b(e)433 b(the)g(pro)72 b(jection)434 b(on)-36 b(to)434 b([)p Fn(u;)221 b Fm(1)p Fw(\).)580 b(Then)433 b(for)h(an)-36 b(y)434 b Fn(u)369 b(>)g Fw(0)433 b(one)h(has)5864 64170 y Fj(\000)6473 65246 y Fn(P)7311 65445 y Fs(u)7911 65246 y Fn(K)9018 65445 y Fs(m;d)10651 65246 y Fn(P)11489 65445 y Fs(u)12090 64170 y Fj(\001)12698 65246 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))370 b(=)e(\002\()p Fn(x)295 b Fm(\000)g Fn(u)p Fw(\))22810 63438 y Fj(Z)23549 66445 y Fk(R)24336 66564 y Ff(\000)25302 65246 y Fw(d)p Fn(w)36 b(L)p Fw(\()p Fn(x;)221 b(w)36 b Fw(\))p Fn(R)11 b Fw(\()p Fn(w)36 b(;)221 b(y)48 b Fw(\)\002\()p Fn(y)341 b Fm(\000)295 b Fn(u)p Fw(\))p Fn(:)2433 b Fw(\(3.21\))5383 68632 y(As)629 b(explained)h(in)g(App)36 b(endix)629 b(C,)679 b(the)630 b(regular)f(part)h(of)g Fn(K)36202 68831 y Fs(m;d)38465 68632 y Fw(is)g(a)g(similarit)-36 b(y)3432 70237 y(transformed)354 b(Laguerre)h(k)-36 b(ernel.)552 b(Hence)355 b(w)-36 b(e)355 b(refer)g(to)g Fn(K)31451 70436 y Fs(m;d)33439 70237 y Fw(also)h(as)f(Laguerre)g(k)-36 b(ernel.)3432 71842 y(It)433 b(has)h(the)f(follo)-36 b(wing)435 b(prop)36 b(erties,)434 b(whic)-36 b(h)433 b(are)h(pro)-36 b(v)g(ed)433 b(in)h(App)36 b(endix)432 b(B.)24367 75163 y(14)p eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 3432 7638 a Fh(Prop)42 b(osition)397 b(3.2.)485 b Fo(L)-66 b(et)382 b Fn(u)369 b(>)g Fw(0)p Fo(.)570 b(Then)382 b Fm(k)p Fn(P)25289 7837 y Fs(u)25890 7638 y Fn(K)26997 7837 y Fs(m;d)28629 7638 y Fn(P)29467 7837 y Fs(u)30068 7638 y Fm(k)369 b Fn(<)g Fw(1)p Fo(.)570 b(In)382 b(addition,)397 b(for)383 b Fn(a)369 b(>)f Fw(0)p Fo(,)8017 10272 y Fn(P)8855 10471 y Fs(u)9456 10272 y Fw(\()p Fl(1)298 b Fm(\000)d Fn(K)13468 10471 y Fs(m;d)15101 10272 y Fw(\))p Fn( )16453 10471 y Fs(a)17377 10272 y Fm(2)368 b Fn(L)19516 9724 y Fq(2)20042 10272 y Fw(\()p Fl(R)p Fw(\))p Fn(;)1553 b(P)24886 10471 y Fs(u)25486 10272 y Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)29499 10471 y Fs(m;d)31132 10272 y Fw(\))31638 9724 y Fi(\003)32164 10272 y Fn( )33010 10471 y Fs(a)33933 10272 y Fm(2)369 b Fn(L)36073 9724 y Fq(2)36599 10272 y Fw(\()p Fl(R)p Fw(\))4588 b(\(3.22\))3432 12906 y Fo(with)464 b(a)h(norm)g(uniformly)f (b)-66 b(ounde)g(d)464 b(in)g Fn(u)p Fo(.)5383 15362 y Fw(With)601 b(these)f(preparations)h(w)-36 b(e)601 b(state)g(the)f(relation)i(b)36 b(et)-36 b(w)g(een)600 b(the)g(equal)i(time)3432 16967 y(k)-36 b(ernel)433 b(of)i(\(3.8\))f (and)f(the)g(Laguerre)g(k)-36 b(ernel.)3432 19422 y Fh(Prop)42 b(osition)452 b(3.3.)522 b Fo(L)-66 b(et)426 b Fw(0)369 b Fn(<)g(a)p Fo(,)433 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Fw(\(4.8\))3432 53707 y(with)433 b Fm(h\001)p Fn(;)221 b Fm(\001i)434 b Fw(denoting)f(the)g(inner)g(pro)36 b(duct)432 b(in)i Fn(L)27373 53225 y Fq(2)27898 53707 y Fw(\()p Fl(R)p Fw(\).)5383 55312 y(W)-108 b(e)433 b(also)i(de\014ne) 17338 56917 y Fn(F)181 b Fw(\()p Fn(u)p Fw(\))368 b(=)h(det)o(\()p Fl(1)299 b Fm(\000)c Fn(P)27412 57116 y Fs(u)28013 56917 y Fn(K)29120 57116 y Fs(m;d)30752 56917 y Fn(P)31590 57116 y Fs(u)32191 56917 y Fw(\))11232 b(\(4.9\))3432 58942 y(and)523 b(supply)g(the)h Fn(m;)221 b(d)524 b Fw(dep)36 b(endence)522 b(of)i Fn(G)25214 58460 y Fs(a;b)26433 58942 y Fw(\()p Fn(u)p Fw(\))g(and)f(of)i Fn(F)181 b Fw(\()p Fn(u)p Fw(\))523 b(when)h(needed.)848 b(W)-108 b(e)3432 60547 y(summarize)433 b(as)3432 62659 y Fh(Prop)42 b(osition)515 b(4.1.)561 b Fo(L)-66 b(et)477 b Fn(w)17592 62858 y Fs(a;b)18811 62659 y Fw(\()p Fn(i;)221 b(j)75 b Fw(\))477 b Fo(b)-66 b(e)477 b(as)h(in)f(De\014nition)e(3.1,)480 b(exc)-66 b(ept)476 b(for)h Fn(w)42489 62858 y Fs(a;b)43708 62659 y Fw(\(0)p Fn(;)221 b Fw(0\))3432 64265 y Fo(for)446 b(which)i(we)f(set)g Fn(w)13971 64464 y Fs(a;b)15190 64265 y Fw(\(0)p Fn(;)221 b Fw(0\))370 b(=)f(0)p Fo(.)591 b(L)-66 b(et)446 b Fn(h)24449 64464 y Fq(0)25422 64265 y Fo(b)-66 b(e)447 b(the)f(c)-66 b(orr)g(esp)g(onding)445 b(top)j(line)e(as)i(given)3432 65870 y(in)464 b(\(3.3\),)h(\(3.4\).)597 b(Then)465 b(for)f Fn(u)370 b(>)e Fw(0)5728 68762 y Fl(P)6741 68132 y Fs(a;b)6741 69088 y Fq(0)7958 68762 y Fw(\()p Fm(f)p Fn(h)9877 68961 y Fq(0)10402 68762 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))370 b Fm(\024)f Fn(u)p Fm(g)p Fw(\))h(=)19184 67863 y(d)p 18814 68457 1463 54 v 18814 69673 a(d)p Fn(u)20410 67686 y Fj(\000)21019 68762 y Fn(F)181 b Fw(\()p Fn(u)p Fw(\))p Fn(G)24820 68214 y Fs(a;b)26038 68762 y Fw(\()p Fn(u)p Fw(\))27790 67686 y Fj(\001)28694 68762 y Fw(+)295 b Fn(F)181 b Fw(\()p Fn(u)p Fw(\)\()p Fn(a)294 b Fw(+)h Fn(b)p Fw(\))p Fn(G)37651 68214 y Fs(a;b)38870 68762 y Fw(\()p Fn(u)p Fw(\))p Fn(;)2296 b Fw(\(4.10\))3432 71510 y Fo(wher)-66 b(e)464 b Fn(F)181 b Fw(\()p Fn(u)p Fw(\))465 b Fo(is)f(given)g(in)g(\(4.9\))h(and)g Fn(G)23521 71028 y Fs(a;b)24740 71510 y Fw(\()p Fn(u)p Fw(\))g Fo(in)f(\(4.8\).) 24367 75163 y Fw(16)p eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 3432 7638 a Fp(5)2151 b(Analytic)717 b(con)-60 b(tin)g(uation)3432 10559 y Fw(W)-108 b(e)494 b(ha)-36 b(v)g(e)494 b(to)g(extend)g(the)f(v)-72 b(alidit)-36 b(y)495 b(of)g(\(4.10\))g(from)f(0)472 b Fn(<)g(a;)221 b(b)472 b(<)g Fw(1)p Fn(=)p Fw(2)495 b(to)f Fn(a)336 b Fw(+)g Fn(b)472 b Fw(=)f(0,)3432 12164 y(whic)-36 b(h)433 b(will)i(b)36 b(e)433 b(ac)-36 b(hiev)g(ed)434 b(b)-36 b(y)433 b(pro)-36 b(ving)434 b(that)f(b)36 b(oth)433 b(sides)h(of)g(\(4.10\))h(are)e(analytic.)3432 14876 y Fh(Prop)42 b(osition)460 b(5.1.)528 b Fo(The)434 b(map)f Fw(\()p Fn(a;)221 b(b)p Fw(\))369 b Fm(7!)h Fl(P)25678 14246 y Fs(a;b)25678 15201 y Fq(0)26895 14876 y Fw(\()p Fm(f)p Fn(h)28814 15075 y Fq(0)29339 14876 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))370 b Fm(\024)g Fn(u)p Fm(g)p Fw(\))433 b Fo(is)h(r)-66 b(e)g(al)433 b(analytic)f(for)3432 16481 y Fn(a;)221 b(b)369 b(>)g Fm(\000)p Fw(1)p Fn(=)p Fw(2)p Fo(.)3432 19193 y(Pr)-66 b(o)g(of.)649 b Fn(h)8253 19392 y Fq(0)8778 19193 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))671 b(is)e(measurable)g(with)h(resp)36 b(ect)669 b(to)g(the)g Fn(\033)48 b Fw(-algebra)669 b(generated)g(b)-36 b(y)3432 20798 y Fn(w)36 b Fw(\()p Fn(i;)221 b Fw(0\),)433 b Fn(w)36 b Fw(\(0)p Fn(;)221 b(j)75 b Fw(\),)434 b Fn(i;)221 b(j)444 b Fw(=)369 b(1)p Fn(;)221 b(:)g(:)g(:)j(;)d(\034)148 b Fw(.)579 b(Let)8605 23731 y Fn(V)9363 23930 y Fs(u)9963 23731 y Fw(\()p Fn(\030)11037 23930 y Fq(1)11563 23731 y Fn(;)221 b(:)g(:)g(:)i(;)e(\030)15043 23930 y Fs(\034)15619 23731 y Fn(;)g(\021)16843 23930 y Fq(1)17370 23731 y Fn(;)g(:)g(:)g(:)i(;)e(\021)20924 23930 y Fs(\034)21500 23731 y Fw(\))8605 25669 y(=)368 b Fl(P)10998 25039 y Fs(a;b)10998 25994 y Fq(0)12215 24593 y Fj(\000)12824 25669 y Fm(f)p Fn(h)14237 25868 y Fq(0)14763 25669 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))370 b Fm(\024)f Fn(u)p Fm(gj)p Fn(w)36 b Fw(\()p Fn(i;)221 b Fw(0\))369 b(=)g Fn(\030)27127 25868 y Fs(i)27503 25669 y Fn(;)221 b(w)36 b Fw(\(0)p Fn(;)221 b(j)75 b Fw(\))369 b(=)g Fn(\021)34300 25868 y Fs(j)34787 25669 y Fn(;)221 b(i;)g(j)444 b Fw(=)369 b(1)p Fn(;)221 b(:)g(:)g(:)j(;)d(\034)43034 24593 y Fj(\001)43929 25669 y Fw(\(5.1\))3432 28602 y(as)297 b(conditional)g(probabilit)-36 b(y)-108 b(.)533 b(Clearly)299 b Fn(V)23613 28801 y Fs(u)24510 28602 y Fw(do)36 b(es)297 b(not)g(dep)36 b(end)296 b(on)g Fn(a;)221 b(b)298 b Fw(and)e(0)370 b Fm(\024)f Fn(V)43220 28801 y Fs(u)44189 28602 y Fm(\024)g Fw(1.)3432 30207 y(Then)3432 33821 y Fl(P)4445 33191 y Fs(a;b)4445 34146 y Fq(0)5662 33821 y Fw(\()p Fm(f)p Fn(h)7581 34020 y Fq(0)8106 33821 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))370 b Fm(\024)f Fn(u)p Fm(g)p Fw(\))1118 b(=)17883 32013 y Fj(Z)18621 35020 y Fk(R)19408 34715 y Fg(2)p Fe(\034)19408 35299 y Fg(+)21201 32160 y Fs(\034)20613 32559 y Fj(Y)20603 35388 y Fs(k)24 b Fq(=1)22541 32745 y Fj(\000)23149 33821 y Fw(d)p Fn(\030)24440 34020 y Fs(k)25009 33821 y Fw(\()25648 33298 y Fq(1)p 25648 33515 471 54 v 25648 34279 a(2)26546 33821 y Fw(+)295 b Fn(a)p Fw(\))p Fn(e)29645 33272 y Fi(\000)p Fq(\(1+2)p Fs(a)p Fq(\))p Fs(\030)33695 33428 y Fe(k)34202 33272 y Fs(=)p Fq(2)35199 32745 y Fj(\001)44081 33821 y Fw(\(5.2\))15743 38138 y Fm(\002)18481 36477 y Fs(\034)17892 36876 y Fj(Y)17883 39705 y Fs(k)24 b Fq(=1)19820 37062 y Fj(\000)20429 38138 y Fw(d)p Fn(\021)21794 38337 y Fs(k)22362 38138 y Fw(\()23001 37615 y Fq(1)p 23001 37832 V 23001 38596 a(2)23899 38138 y Fw(+)295 b Fn(b)p Fw(\))p Fn(e)26868 37589 y Fi(\000)p Fq(\(1+2)p Fs(b)p Fq(\))p Fs(\021)30879 37745 y Fe(k)31387 37589 y Fs(=)p Fq(2)32383 37062 y Fj(\001)32992 38138 y Fn(V)33750 38337 y Fs(u)34350 38138 y Fw(\()p Fn(\030)35424 38337 y Fq(1)35950 38138 y Fn(;)221 b(:)g(:)g(:)j(;)d(\030)39431 38337 y Fs(\034)40007 38138 y Fn(;)g(\021)41231 38337 y Fq(1)41757 38138 y Fn(;)g(:)g(:)g(:)j(;)d(\021)45312 38337 y Fs(\034)45887 38138 y Fw(\))p Fn(;)3432 42273 y Fw(whic)-36 b(h)433 b(b)-36 b(y)433 b(insp)36 b(ection)434 b(is)g(real)g(analytic)g(for)h Fn(a;)221 b(b)369 b(>)g Fm(\000)p Fw(1)p Fn(=)p Fw(2.)p 45663 42273 45 878 v 45708 41439 781 45 v 45708 42273 V 46487 42273 45 878 v 3432 44985 a Fh(Prop)42 b(osition)410 b(5.2.)494 b Fo(L)-66 b(et)393 b Fn(u)369 b(>)g Fw(0)393 b Fo(and)h(let)f Fn(G)25173 44503 y Fs(a;b)26392 44985 y Fw(\()p Fn(u)p Fw(\))g Fo(b)-66 b(e)393 b(given)f(by)h(\(4.8\).)574 b(Then)393 b Fw(\()p Fn(a;)221 b(b)p Fw(\))369 b Fm(7!)3432 46590 y Fn(G)4458 46108 y Fs(a;b)5677 46590 y Fw(\()p Fn(u)p Fw(\))464 b Fo(extends)g(to)h(a)g(r)-66 b(e)g(al)464 b(analytic)g(function)g(for)g Fn(a;)221 b(b)370 b Fm(2)e Fw(\()p Fm(\000)p Fw(1)p Fn(=)p Fw(2)p Fn(;)221 b Fw(1)p Fn(=)p Fw(2\))p Fo(.)3432 49302 y(Pr)-66 b(o)g(of.)649 b Fw(W)-108 b(e)433 b(rep)36 b(eat)434 b(Eq.)g(\(4.8\),)3432 52236 y(\()p Fn(a)90 b Fw(+)g Fn(b)p Fw(\))p Fn(Z)7762 52435 y Fs(a;b)8980 52236 y Fn(G)10006 51687 y Fs(a;b)11225 52236 y Fw(\()p Fn(u)p Fw(\))369 b(=)f Fn(Z)15616 52435 y Fs(a;b)16926 52236 y Fm(\000)90 b(h)p Fn( )19412 52435 y Fs(a)19966 52236 y Fn(;)221 b Fw(\()p Fl(1)90 b Fm(\000)g Fn(K)24147 52435 y Fs(m;d)25784 52236 y Fw(\))p Fn(P)27128 52435 y Fs(u)27728 52236 y Fw(\()p Fl(1)g Fm(\000)g Fn(P)31058 52435 y Fs(u)31662 52236 y Fn(K)32769 52435 y Fs(m;d)34402 52236 y Fn(P)35240 52435 y Fs(u)35840 52236 y Fw(\))36346 51687 y Fi(\000)p Fq(1)37604 52236 y Fn(P)38442 52435 y Fs(u)39042 52236 y Fw(\()p Fl(1)g Fm(\000)g Fn(K)42641 52435 y Fs(m;d)44277 52236 y Fw(\))p Fn( )45629 52435 y Fs(b)46086 52236 y Fm(i)43929 53841 y Fw(\(5.3\))3432 55446 y(with)433 b Fn( )7240 55645 y Fs(a)7795 55446 y Fw(\()p Fn(x)p Fw(\))369 b(=)f Fn(e)11898 54964 y Fi(\000)p Fs(ax)13715 55446 y Fw(.)5383 57051 y(First)495 b(remark)h(that)f(\()p Fn(a)337 b Fw(+)g Fn(b)p Fw(\))p Fn(Z)20823 57250 y Fs(a;b)22538 57051 y Fw(is)496 b(real)g(analytic)h(for)f Fn(a;)221 b(b)475 b Fm(2)g Fw(\()p Fm(\000)p Fw(1)p Fn(=)p Fw(2)p Fn(;)221 b Fw(1)p Fn(=)p Fw(2\).)767 b(On)3432 58656 y(the)598 b(other)h(hand,)640 b(in)599 b(\(5.3\))g(the)g(\014rst)f (\(th)-36 b(us)598 b(also)i(the)f(second\))f(term)h(div)-36 b(erges)600 b(as)3432 60261 y Fn(a)353 b Fw(+)h Fn(b)516 b Fm(!)g Fw(0.)838 b(Th)-36 b(us)519 b(w)-36 b(e)520 b(ha)-36 b(v)g(e)520 b(to)g(\014nd)f(another)g(represen)-36 b(tation)519 b(of)i Fn(G)39460 59779 y Fs(a;b)41198 60261 y Fw(suc)-36 b(h)519 b(that)3432 61866 y(b)36 b(oth)393 b(terms)h(remain)h(\014nite)e(in)i(the)f Fn(a)215 b Fw(+)g Fn(b)368 b Fm(!)i Fw(0)394 b(limit.)566 b(F)-108 b(rom)394 b(eac)-36 b(h)394 b(term)g(w)-36 b(e)395 b(subtract)3432 63471 y(the)433 b(quan)-36 b(tit)g(y)433 b Fm(h)p Fn( )12237 63670 y Fs(a)12792 63471 y Fn(;)221 b(P)14212 63670 y Fs(u)14813 63471 y Fw(\()p Fl(1)298 b Fm(\000)d Fn(K)18825 63670 y Fs(m;d)20458 63471 y Fw(\))p Fn( )21810 63670 y Fs(b)22267 63471 y Fm(i)434 b Fw(and)f(obtain)3473 66405 y(r.h.s.)h(of)g(\(5)p Fn(:)p Fw(3\))1108 b(=)14113 65329 y Fj(\000)14721 66405 y Fn(Z)15611 66604 y Fs(a;b)17126 66405 y Fm(\000)295 b(h)p Fn( )19817 66604 y Fs(a)20371 66405 y Fn(;)221 b(P)21791 66604 y Fs(u)22393 66405 y Fn( )23239 66604 y Fs(b)23696 66405 y Fm(i)24213 65329 y Fj(\001)25117 66405 y Fw(+)294 b Fm(h)p Fn( )27786 66604 y Fs(a)28341 66405 y Fn(;)221 b(P)29761 66604 y Fs(u)30362 66405 y Fn(K)31469 66604 y Fs(m;d)33102 66405 y Fn( )33948 66604 y Fs(b)34405 66405 y Fm(i)9007 b Fw(\(5.4\))14113 68342 y Fm(\000h)p Fn( )16509 68541 y Fs(a)17063 68342 y Fn(;)221 b(Q)18675 68541 y Fs(u)19276 68342 y Fw(\()p Fl(1)298 b Fm(\000)e Fn(K)23289 68541 y Fs(m;d)24921 68342 y Fw(\))p Fn(P)26265 68541 y Fs(u)26866 68342 y Fw(\()p Fl(1)i Fm(\000)d Fn(P)30609 68541 y Fs(u)31210 68342 y Fn(K)32317 68541 y Fs(m;d)33950 68342 y Fn(P)34788 68541 y Fs(u)35388 68342 y Fw(\))35894 67793 y Fi(\000)p Fq(1)37152 68342 y Fn(P)37990 68541 y Fs(u)38590 68342 y Fw(\()p Fl(1)j Fm(\000)e Fn(K)42603 68541 y Fs(m;d)44236 68342 y Fw(\))p Fn( )45588 68541 y Fs(b)46045 68342 y Fm(i)24367 75163 y Fw(17)p eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 3432 7638 a Fw(where)411 b Fn(Q)8197 7837 y Fs(u)9167 7638 y Fw(=)368 b Fl(1)254 b Fm(\000)d Fn(P)13696 7837 y Fs(u)14297 7638 y Fw(.)571 b(With)412 b(this)f(rearrangemen)-36 b(t)412 b(one)f(singles)i(out)e (the)h(div)-36 b(ergence)3432 9243 y(whic)g(h)387 b(no)-36 b(w)388 b(are)f(in)h Fn(Z)14254 9442 y Fs(a;b)15860 9243 y Fw(and)f Fm(h)p Fn( )19706 9442 y Fs(a)20261 9243 y Fn(;)221 b(P)21681 9442 y Fs(u)22282 9243 y Fn( )23128 9442 y Fs(b)23585 9243 y Fm(i)387 b Fw(only)-108 b(.)564 b(W)-108 b(e)387 b(discuss)g(the)g(analytic)i(con)-36 b(tin)g(ua-)3432 10848 y(tion)445 b(from)h Fn(a;)221 b(b)389 b Fm(2)f Fw(\(0)p Fn(;)221 b Fw(1)p Fn(=)p Fw(2\))447 b(to)f Fn(a;)221 b(b)389 b Fm(2)g Fw(\()p Fm(\000)p Fw(1)p Fn(=)p Fw(2)p Fn(;)221 b Fw(1)p Fn(=)p Fw(2\))447 b(for)f(the)f(three)f (terms)h(separately)-108 b(,)3432 12453 y(where)433 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49588 y Fw(as)375 b Fn(\014)31507 49787 y Fq(1)32402 49588 y Fm(!)369 b Fw(1)176 b Fm(\000)g Fn(\032)p Fw(,)387 b Fn(\014)38287 49787 y Fq(2)38989 49588 y Fw(+)40309 49065 y Fq(1)p 40309 49282 V 40309 50046 a(2)41088 49588 y Fm(\000)176 b Fn(\032)g Fm(\000)g Fn(b)369 b Fm(!)3564 50670 y Fq(1)p 3564 50888 V 3564 51651 a(2)4463 51193 y Fm(\000)295 b Fn(b)369 b(>)g Fw(0)434 b(as)g Fn(\014)11509 51392 y Fq(2)12404 51193 y Fm(!)369 b Fn(\032)p Fw(,)434 b(and)f Fn(\014)18830 51392 y Fq(1)19651 51193 y Fw(+)295 b Fn(\014)21692 51392 y Fq(2)22218 51193 y Fw(.)578 b(It)434 b(follo)-36 b(ws)13845 53699 y(~)-696 b Fn(g)14422 53898 y Fs(b)15249 53699 y Fm(2)368 b Fn(L)17388 53150 y Fq(2)17914 53699 y Fw(\(\(0)p Fn(;)221 b Fm(1)p Fw(\))p Fn(;)g(e)23177 53150 y Fs(\026x)24329 53699 y Fw(d)p Fn(x)p Fw(\))1301 b(for)434 b(all)g Fn(\026)369 b(<)f Fw(1)p Fn(=)p Fw(2)p Fn(:)7045 b Fw(\(5.17\))3432 56204 y(F)-108 b(rom)335 b(this)g(one)h(deduces)e(that)i Fm(h)p Fn(f)20102 56403 y Fs(a)20656 56204 y 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Fs(a)32489 37723 y Fn(;)221 b(K)34178 37922 y Fs(m;d)35812 37723 y Fn(P)36650 37922 y Fs(u)37250 37723 y Fn( )38096 37922 y Fi(\000)p Fs(a)39383 37723 y Fm(i)3379 b Fw(\(5.22\))11462 40691 y Fm(\000h)p Fn( )13858 40890 y Fs(a)14413 40691 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)19008 40890 y Fs(m;d)20641 40691 y Fw(\))p Fn(P)21985 40890 y Fs(u)22585 40691 y Fw(\()p Fl(1)k Fm(\000)c Fn(P)26329 40890 y Fs(u)26929 40691 y Fn(K)28036 40890 y Fs(m;d)29669 40691 y Fn(P)30507 40890 y Fs(u)31108 40691 y Fw(\))31614 40143 y Fi(\000)p Fq(1)32871 40691 y Fn(P)33709 40890 y Fs(u)34310 40691 y Fw(\()p Fl(1)j Fm(\000)d Fn(K)38322 40890 y Fs(m;d)39955 40691 y Fw(\))p Fn(Q)41491 40890 y Fs(u)42091 40691 y Fn( )42937 40890 y Fi(\000)p Fs(a)44224 40691 y Fm(i)p Fn(:)5383 43624 y Fw(With)433 b(the)g(same)h(con)-36 b(v)g(en)g(tion,)434 b(let)f(us)h(denote)21064 46558 y(lim)20680 47394 y Fs(b)p Fi(!\000)p Fs(a)23477 46558 y Fl(P)24490 45928 y Fs(a;b)24490 46884 y Fq(0)26076 46558 y Fw(=)368 b Fl(P)28469 46757 y Fq(0)28993 46558 y Fn(:)13925 b Fw(\(5.23\))3432 49981 y(Recall)632 b(that)f(b)-36 b(y)632 b(construction)f Fl(P)21231 50180 y Fq(0)22386 49981 y Fw(is)h(the)f(probabilit)-36 b(y)632 b(measure)f(for)h(the)f (family)3432 51586 y Fn(w)36 b Fw(\()p Fn(i;)221 b(j)75 b Fw(\),)473 b Fn(i;)221 b(j)498 b Fm(2)423 b Fl(N)p Fw(,)468 b(of)e(indep)36 b(enden)-36 b(t)463 b(exp)36 b(onen)-36 b(tially)467 b(distributed)d(random)h(v)-72 b(ariables)3432 53191 y(suc)-36 b(h)404 b(that)h Fn(w)36 b Fw(\(0)p Fn(;)221 b Fw(0\))369 b(=)g(0,)411 b Fn(w)36 b Fw(\()p Fn(i;)221 b Fw(0\))405 b(has)g(mean)g(\(1)238 b Fm(\000)f Fn(\032)p Fw(\))29790 52709 y Fi(\000)p Fq(1)31453 53191 y Fw(for)405 b Fn(i)369 b Fm(\025)g Fw(1,)412 b Fn(w)36 b Fw(\(0)p Fn(;)221 b(j)75 b Fw(\))405 b(has)g(mean)3432 54796 y Fn(\032)4103 54314 y Fi(\000)p Fq(1)5902 54796 y Fw(for)542 b Fn(j)627 b Fm(\025)553 b Fw(1,)569 b(and)541 b(otherwise)h Fn(w)36 b Fw(\()p Fn(i;)221 b(j)75 b Fw(\))541 b(has)g(mean)h(1.)902 b(This)542 b(is)g(precisely)g(the)f Fn(w)36 b Fw(-)3432 56401 y(marginal)437 b(of)h Fl(Q)j Fw(in)c(Section)f(2.)588 b(Also,)439 b(b)-36 b(y)436 b(de\014nition,)h Fn(h)31591 55919 y Fq(0)32117 56401 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))375 b(=)f Fn(h)37474 56600 y Fq(0)38000 56401 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))438 b(path)-36 b(wise.)3432 58006 y(Th)g(us)433 b(one)g(concludes)3432 60718 y Fh(Prop)42 b(osition)499 b(5.3.)554 b Fo(F)-100 b(or)466 b Fw(0)369 b Fn(<)g(c)19769 60917 y Fq(1)20663 60718 y Fn(<)g(c)22604 60917 y Fq(2)23499 60718 y Fn(<)f Fm(1)465 b Fo(it)g(holds)8516 62446 y Fj(Z)9844 62797 y Fs(c)10252 62920 y Fg(2)9254 65454 y Fs(c)9662 65577 y Fg(1)10990 64254 y Fw(d)p Fn(u)p Fl(Q)13586 63178 y Fj(\000)14199 64254 y Fm(f)p Fn(h)15612 63706 y Fq(0)16138 64254 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))370 b Fm(\024)f Fn(u)p Fm(g)22161 63178 y Fj(\001)23877 64254 y Fw(=)25996 62446 y Fj(Z)27324 62797 y Fs(c)27732 62920 y Fg(2)26734 65454 y Fs(c)27142 65577 y 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24820 y Fq(1)p Fs(=)p Fq(3)28181 25368 y Fw(+)295 b Fm(O)37 b Fw(\(1\))11034 b(\(6.13\))3432 28297 y(with)19546 29902 y Fn(\024)368 b Fw(=)h Fn(\032)22715 29354 y Fi(\000)p Fq(1)23973 29902 y Fw(\(1)295 b Fm(\000)g Fn(\032)p Fw(\))27929 29354 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)30128 29902 y Fn(:)12790 b Fw(\(6.14\))5383 32225 y(After)376 b(edge)g(scaling)h(the)f(terms)f (of)i Fn(G)24034 32424 y Fq(0)24936 32225 y Fw(will)g(b)36 b(e)376 b(expressed)g(via)h(the)f(functions)g Fn(H)45707 32424 y Fs(N)3432 33830 y Fw(and)6298 33494 y(~)5961 33830 y Fn(H)7039 34029 y Fs(N)7935 33830 y Fw(,)434 b(de\014ned)e(as)11098 37567 y Fn(H)12176 37766 y Fs(N)13072 37567 y Fw(\()p Fn(y)48 b Fw(\))1106 b(=)h Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))20804 36669 y Fn(\024N)22734 36187 y Fq(1)p Fs(=)p Fq(3)p 20804 37262 3396 54 v 21045 38478 a Fm(\000)p Fw(2)p Fn(\031)48 b(i)24332 37567 y(I)24906 37774 y Fs(N)94 b Fq(+)p Fs(\013=)p Fq(2)p Fs(;\013=)p Fq(2)29886 37567 y Fw(\()p Fn(u)296 b Fw(+)e Fn(y)48 b(\024N)35346 37019 y Fq(1)p Fs(=)p Fq(3)36812 37567 y Fw(\))p Fn(;)5600 b Fw(\(6.15\))11435 40741 y(~)11098 41077 y Fn(H)12176 41276 y Fs(N)13072 41077 y Fw(\()p Fn(y)48 b Fw(\))1106 b(=)h Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))20671 40528 y Fi(\000)p Fq(1)22062 40178 y Fn(\024N)23992 39696 y Fq(1)p Fs(=)p Fq(3)p 22062 40771 V 22820 41988 a Fw(2)p Fn(\031)48 b(i)25749 40741 y Fw(~)25590 41077 y Fn(I)26164 41283 y Fs(N)94 b Fq(+)p Fs(\013=)p Fq(2)p Fs(;\013=)p Fq(2)31144 41077 y Fw(\()p Fn(u)295 b Fw(+)g Fn(y)48 b(\024N)36604 40528 y Fq(1)p Fs(=)p Fq(3)38070 41077 y Fw(\))p Fn(:)3432 44640 y Fw(Using)433 b(the)g(b)36 b(ounds)433 b(on)g Fn(I)538 b Fw(and)19456 44304 y(~)19298 44640 y Fn(I)g Fw(of)434 b(Section)f(7)h(w)-36 b(e)434 b(obtain,)g(for)g(an)-36 b(y)434 b Fn(\014)443 b(>)368 b Fw(0)434 b(\014xed,)18890 47568 y Fm(j)p Fn(H)20337 47767 y Fs(N)21233 47568 y Fw(\()p Fn(y)48 b Fw(\))p Fm(j)1106 b(\024)h Fn(C)27473 47767 y Fs(\014)28102 47568 y Fn(e)28705 47020 y Fi(\000)p Fs(\014)46 b(y)30562 47568 y Fn(;)12356 b Fw(\(6.16\))18890 49506 y Fm(j)19596 49170 y Fw(~)19259 49506 y Fn(H)20337 49705 y Fs(N)21233 49506 y Fw(\()p Fn(y)48 b Fw(\))p Fm(j)1106 b(\024)h Fn(C)27473 49705 y Fs(\014)28102 49506 y Fn(e)28705 48957 y Fi(\000)p Fs(\014)46 b(y)3432 52434 y Fw(for)504 b(some)g Fn(C)9749 52633 y Fs(\014)10865 52434 y Fn(>)488 b Fw(0)504 b(indep)36 b(enden)-36 b(t)502 b(of)i Fn(N)642 b Fw(and)503 b Fn(y)537 b Fm(\025)488 b Fw(0.)789 b(Moreo)-36 b(v)g(er)504 b(w)-36 b(e)504 b(also)h(ha)-36 b(v)g(e)504 b(the)3432 54039 y(p)36 b(oin)-36 b(t)g(wise)433 b(con)-36 b(v)g(ergence)8535 56968 y(lim)8077 57795 y Fs(N)94 b Fi(!1)11021 56968 y Fn(H)12099 57167 y Fs(N)12995 56968 y Fw(\()p Fn(y)48 b Fw(\))1107 b(=)f(Ai)q(\()p Fn(w)20723 56420 y Fq(2)21543 56968 y Fw(+)295 b Fn(s)g Fw(+)f Fn(y)48 b Fw(\))p Fn(e)26855 56420 y Fs(w)24 b Fq(\()p Fs(w)28619 56107 y Fg(2)29080 56420 y Fq(+)p Fs(s)p Fq(+)p Fs(y)32 b Fq(\))31897 56968 y Fn(e)32500 56420 y Fi(\000)33365 56057 y Fg(1)p 33364 56218 406 40 v 33364 56767 a(3)33903 56420 y Fs(w)34602 56107 y Fg(3)35488 56968 y Fw(=)368 b Fn(')37720 57167 y Fs(w)24 b(;s)39171 56968 y Fw(\()p Fn(y)48 b Fw(\))p Fn(;)2053 b Fw(\(6.17\))8535 59709 y(lim)8077 60535 y Fs(N)94 b Fi(!1)11358 59373 y Fw(~)11021 59709 y Fn(H)12099 59908 y Fs(N)12995 59709 y Fw(\()p Fn(y)48 b Fw(\))1107 b(=)f(Ai)q(\()p Fn(w)20723 59160 y Fq(2)21543 59709 y Fw(+)295 b Fn(s)g Fw(+)f Fn(y)48 b Fw(\))p Fn(e)26855 59160 y Fi(\000)p Fs(w)24 b Fq(\()p Fs(w)29351 58848 y Fg(2)29812 59160 y Fq(+)p Fs(s)p Fq(+)p Fs(y)32 b Fq(\))32629 59709 y Fn(e)33365 58798 y Fg(1)p 33364 58959 V 33364 59508 a(3)33903 59160 y Fs(w)34602 58848 y Fg(3)35488 59709 y Fw(=)368 b Fn(')37720 59908 y Fi(\000)p Fs(w)24 b(;s)39903 59709 y Fw(\()p Fn(y)48 b Fw(\))p Fn(:)3432 63025 y Fw(W)-108 b(e)433 b(simplify)i(the)e(notations)h(in)f(this)g (pro)36 b(of)435 b(b)-36 b(y)433 b(setting)h Fn(K)464 b Fw(=)368 b Fn(K)36133 63231 y Fs(N)94 b Fq(+)p Fs(\013=)p Fq(2)p Fs(;\013=)p Fq(2)41113 63025 y Fw(.)3432 65294 y Fh(The)498 b(estimate)i(of)e(the)g(terms)h(for)g Fn(\024)24043 64812 y Fi(\000)p Fq(1)25300 65294 y Fn(N)26481 64812 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)28679 65294 y Fn(G)29705 65493 y Fq(0)30231 65294 y Fw(\()p Fn(u)295 b Fw(+)g Fn(s\024N)35622 64812 y Fq(1)p Fs(=)p Fq(3)37087 65294 y Fw(\).)3432 67563 y Fo(First)498 b(term)g(of)g(\(5.21\))h(and)g (\(5.22\).)689 b Fw(Using)470 b(\(6.12\),)481 b(2)p Fn(d)432 b Fw(=)f Fn(\013)8 b Fw(,)480 b(and)470 b Fn(m)320 b Fm(\000)g Fn(d)432 b Fw(=)f Fn(N)139 b Fw(,)479 b(w)-36 b(e)3432 69168 y(ha)g(v)g(e)18953 71100 y(lim)18494 71927 y Fs(N)94 b Fi(!1)21439 71100 y Fn(u)295 b Fw(+)23914 70202 y(2)p Fn(ad)g Fm(\000)g Fn(m)p 23914 70795 4771 54 v 24648 71525 a Fq(1)p 24648 71742 471 54 v 24648 72506 a(4)25546 72048 y Fm(\000)g Fn(a)27557 71664 y Fq(2)29186 71100 y Fw(=)368 b Fn(s:)11739 b Fw(\(6.18\))24367 75163 y(23)p eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop 3432 7638 a Fo(Se)-66 b(c)g(ond)463 b(term)h(of)g(\(5.21\))i(for)f Fn(w)404 b Fm(\025)369 b Fw(0)p Fo(.)578 b Fw(W)-108 b(e)434 b(compute)f(the)g (limit)17235 10514 y(lim)16777 11341 y Fs(N)94 b Fi(!1)19721 10514 y Fn(\024)20470 9966 y Fi(\000)p Fq(1)21728 10514 y Fn(N)22909 9966 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)25107 10514 y Fm(h)p Fn( )26470 10713 y Fs(a)27024 10514 y Fn(;)221 b(P)28444 10713 y Fs(u)29045 10514 y Fn(K)95 b( )31093 10713 y Fi(\000)p Fs(a)32380 10514 y Fm(i)p Fn(:)10021 b Fw(\(6.19\))3432 13852 y Fn( )4278 14051 y Fi(\000)p Fs(a)5998 13852 y Fw(is)434 b(eigenfunction)f(of)i Fn(R)11 b Fw(,)433 b(see)h(\(3.15\).)579 b(Th)-36 b(us)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)4400 17405 y Fm(h)p Fn( )5763 17604 y Fs(a)6318 17405 y Fn(;)221 b(P)7738 17604 y Fs(u)8339 17405 y Fn(K)95 b( )10387 17604 y Fi(\000)p Fs(a)11673 17405 y Fm(i)369 b Fw(=)g Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))p Fm(h)p Fn( )17983 17604 y Fs(a)18537 17405 y Fn(;)221 b(P)19957 17604 y Fs(u)20558 17405 y Fn(LP)22281 17604 y Fi(\000)23069 17405 y Fn( 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Fj(Z)26179 25016 y Fi(1)25589 27673 y Fq(0)27397 26474 y Fw(d)p Fn(x)29080 24666 y Fj(Z)30408 25016 y Fi(1)29818 27673 y Fq(0)31626 26474 y Fw(d)p Fn(y)48 b(')33883 26673 y Fs(w)24 b(;s)35333 26474 y Fw(\()p Fn(x)295 b Fw(+)g Fn(y)48 b Fw(\))p Fn(:)3550 b Fw(\(6.21\))3432 30716 y Fo(Se)-66 b(c)g(ond)425 b(term)i(of)g (\(5.22\))h(for)f Fn(w)404 b Fm(\024)369 b Fw(0)p Fo(.)565 b Fw(This)393 b(case)g(is)g(analogous)h(to)f(the)f(previous)h(one.)3432 32321 y(One)432 b(obtains)7074 35541 y(lim)6616 36368 y Fs(N)94 b Fi(!1)9560 35541 y Fn(\024)10309 34993 y Fi(\000)p Fq(1)11567 35541 y Fn(N)12748 34993 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)14945 35541 y Fm(h)p Fn( )16308 35740 y Fs(a)16863 35541 y Fn(;)221 b(K)95 b(P)19485 35740 y Fs(u)20086 35541 y Fn( )20932 35740 y Fi(\000)p Fs(a)22218 35541 y Fm(i)369 b Fw(=)24485 33733 y Fj(Z)25813 34084 y Fi(1)25223 36741 y Fq(0)27031 35541 y Fw(d)p Fn(x)28714 33733 y Fj(Z)30042 34084 y Fi(1)29452 36741 y Fq(0)31260 35541 y Fw(d)p Fn(y)48 b(')33517 35740 y Fi(\000)p Fs(w)24 b(;s)35699 35541 y Fw(\()p Fn(x)295 b Fw(+)g Fn(y)48 b Fw(\))p Fn(:)3184 b Fw(\(6.22\))3432 39119 y(The)506 b(sum)f(of)i(\(6.18\))g(and)f(\(6.21\),)526 b(resp.)506 b(\(6.22\),)525 b(yields)507 b Fn(S)33124 39326 y Fi(j)p Fs(w)24 b Fi(j)p Fs(;s)35604 39119 y Fw(as)506 b(the)g(\014rst)f(term)h (in)3432 40724 y(Theorem)433 b(6.2.)3432 42993 y Fo(Thir)-66 b(d)464 b(term)g(of)g(\(5.21\))i(for)f Fn(w)404 b Fm(\025)369 b Fw(0)p Fo(.)578 b Fw(The)434 b(third)e(term)i(of)g Fn(\024)33288 42511 y Fi(\000)p Fq(1)34545 42993 y Fn(N)35726 42511 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)37924 42993 y Fn(G)38950 43192 y Fq(0)39476 42993 y Fw(\()p Fn(u)p Fw(\))f(is)18709 45870 y Fn(\024)19458 45321 y Fi(\000)p Fq(1)20715 45870 y Fn(N)21896 45321 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)24094 45870 y Fm(h)24756 45534 y Fw(~)24611 45870 y(\010)25550 46069 y Fs(N)26447 45870 y Fn(;)27372 45534 y Fw(~)27029 45870 y Fn(A)28004 46069 y Fs(N)29082 45534 y Fw(~)28901 45870 y(\011)29913 46069 y Fs(N)30809 45870 y Fm(i)11953 b Fw(\(6.23\))3432 48746 y(with)14296 51286 y(~)14152 51622 y(\010)15091 51821 y Fs(N)15988 51622 y Fw(\()p Fn(x)p Fw(\))1106 b(=)h Fm(\000)21997 50546 y Fj(\000)22606 51622 y Fn(K)23808 51074 y Fi(\003)24334 51622 y Fn(Q)25364 51821 y Fs(u)25964 51622 y Fn( )26810 51821 y Fs(a)27365 50546 y Fj(\001)27973 51622 y Fw(\()p Fn(x)p Fw(\))p Fn(;)14317 53412 y Fw(~)14137 53747 y(\011)15149 53946 y Fs(N)16045 53747 y Fw(\()p Fn(y)48 b Fw(\))1106 b(=)20964 52671 y Fj(\000)21573 53747 y Fw(\()p Fl(1)298 b Fm(\000)d Fn(K)95 b Fw(\))p Fn( )27032 53946 y Fi(\000)p Fs(a)28319 52671 y Fj(\001)28928 53747 y Fw(\()p Fn(y)48 b Fw(\))p Fn(;)12296 b Fw(\(6.24\))13194 55537 y(~)12851 55872 y Fn(A)13826 56071 y Fs(N)14723 55872 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))1107 b(=)20964 54797 y Fj(\000)21573 55872 y Fn(P)22411 56071 y Fs(u)23011 55872 y Fw(\()p Fl(1)299 b Fm(\000)c Fn(P)26755 56071 y Fs(u)27355 55872 y Fn(K)95 b(P)29395 56071 y Fs(u)29996 55872 y Fw(\))30502 55324 y Fi(\000)p Fq(1)31759 55872 y Fn(P)32597 56071 y Fs(u)33198 54797 y Fj(\001)33807 55872 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))p Fn(:)3432 58866 y Fw(T)-108 b(o)447 b(establish)h(the)f (scaling)i(limit,)i Fn(x)393 b Fw(=)g Fn(u)305 b Fw(+)f Fn(\030)64 b(\024N)28220 58384 y Fq(1)p Fs(=)p Fq(3)29686 58866 y Fw(,)451 b(w)-36 b(e)448 b(de\014ne)f(the)g(rescaled)g(quan-) 3432 60471 y(tities)9683 63347 y Fn( )10577 62799 y Fs(`)10529 63675 y(a)11084 63347 y Fw(\()p Fn(\030)64 b Fw(\))1107 b(=)f Fn( )16799 63546 y Fs(a)17354 63347 y Fw(\()p Fn(u)295 b Fw(+)g Fn(\030)64 b(\024N)22764 62799 y Fq(1)p Fs(=)p Fq(3)24230 63347 y Fw(\))p Fn(\027)84 b Fw(\()p Fn(\030)64 b Fw(\))27107 62799 y Fi(\000)p Fq(1)28364 63347 y Fn(;)8951 65284 y( )9845 64736 y Fs(r)9797 65613 y Fi(\000)p Fs(a)11084 65284 y Fw(\()p Fn(\030)g Fw(\))1107 b(=)f Fn( )16799 65483 y Fi(\000)p Fs(a)18086 65284 y Fw(\()p Fn(u)295 b Fw(+)g Fn(\030)64 b(\024N)23496 64736 y Fq(1)p Fs(=)p Fq(3)24962 65284 y Fw(\))p Fn(\027)84 b Fw(\()p Fn(\030)64 b Fw(\))p Fn(;)15079 b Fw(\(6.25\))6943 67360 y Fn(K)8145 66812 y Fs(r)8050 67689 y(N)8946 67360 y Fw(\()p Fn(\030)10020 67559 y Fq(1)10545 67360 y Fn(;)221 b(\030)11695 67559 y Fq(2)12222 67360 y Fw(\))1107 b(=)f Fn(\024N)17883 66812 y Fq(1)p Fs(=)p Fq(3)19349 67360 y Fn(K)95 b Fw(\()p Fn(u)295 b Fw(+)g Fn(\030)23967 67559 y Fq(1)24493 67360 y Fn(\024N)26423 66812 y Fq(1)p Fs(=)p Fq(3)27889 67360 y Fn(;)221 b(u)296 b Fw(+)f Fn(\030)31382 67559 y Fq(2)31908 67360 y Fn(\024N)33838 66812 y Fq(1)p Fs(=)p Fq(3)35303 67360 y Fw(\))p Fn(\027)84 b Fw(\()p Fn(\030)37610 67559 y Fq(1)38135 67360 y Fw(\))p Fn(\027)g Fw(\()p Fn(\030)40442 67559 y Fq(2)40967 67360 y Fw(\))41473 66812 y Fi(\000)p Fq(1)42731 67360 y Fn(;)3432 70237 y Fw(where)17997 71842 y Fn(\027)g Fw(\()p Fn(\030)64 b Fw(\))369 b(=)f Fn(e)22720 71293 y Fi(\000)p Fs(a)p Fq(\()p Fs(u)p Fq(+)p Fs(\030)35 b(\024N)27427 70981 y Fg(1)p Fe(=)p Fg(3)28717 71293 y Fq(\))29139 71842 y Fn(e)29742 71293 y Fi(\000)p Fs(w)24 b(\030)31676 71842 y Fn(:)11242 b Fw(\(6.26\))24367 75163 y(24)p eop end end %%Page: 25 25 TeXDict begin HPSdict begin 25 24 bop 3432 7638 a Fw(Then)433 b(\(6.23\))h(b)36 b(ecomes)21402 9243 y Fm(h)p Fw(\010)22858 9442 y Fs(N)23754 9243 y Fn(;)221 b(A)25311 9442 y Fs(N)26208 9243 y Fw(\011)27220 9442 y Fs(N)28116 9243 y Fm(i)14646 b Fw(\(6.27\))3432 11410 y(with)13545 13977 y(\010)14484 14176 y Fs(N)15381 13977 y Fw(\()p Fn(\030)16455 14176 y Fq(1)16981 13977 y Fw(\))1106 b(=)h Fm(\000)21745 12901 y Fj(\000)22354 13977 y Fn(K)23556 13347 y Fs(r)-26 b(;)p Fi(\003)23461 14337 y Fs(N)24741 13977 y Fn(Q)25771 14176 y Fq(0)26297 13977 y Fn( )27191 13428 y Fs(`)27143 14305 y(a)27698 12901 y Fj(\001)28307 13977 y Fw(\()p Fn(\030)29381 14176 y Fq(1)29907 13977 y Fw(\))p Fn(;)13473 15914 y Fw(\011)14485 16113 y Fs(N)15381 15914 y Fw(\()p Fn(\030)16455 16113 y Fq(2)16981 15914 y Fw(\))1106 b(=)20712 14838 y Fj(\000)21321 15914 y Fw(\()p Fl(1)298 b Fm(\000)d Fn(K)25428 15366 y Fs(r)25333 16242 y(N)26230 15914 y Fw(\))p Fn( )27630 15366 y Fs(r)27582 16242 y Fi(\000)p Fs(a)28868 14838 y Fj(\001)29477 15914 y Fw(\()p Fn(\030)30551 16113 y Fq(2)31076 15914 y Fw(\))p Fn(;)11336 b Fw(\(6.28\))11833 17851 y Fn(A)12808 18050 y Fs(N)13704 17851 y Fw(\()p Fn(\030)14778 18050 y Fq(1)15304 17851 y Fn(;)221 b(\030)16454 18050 y Fq(2)16981 17851 y Fw(\))1106 b(=)20712 16775 y Fj(\000)21321 17851 y Fn(P)22159 18050 y Fq(0)22685 17851 y Fw(\()p Fl(1)298 b Fm(\000)e Fn(P)26429 18050 y Fq(0)26955 17851 y Fn(K)28157 17303 y Fs(r)28062 18180 y(N)28958 17851 y Fn(P)29796 18050 y Fq(0)30322 17851 y Fw(\))30828 17303 y Fi(\000)p Fq(1)32086 17851 y Fn(P)32924 18050 y Fq(0)33450 16775 y Fj(\001)34059 17851 y Fw(\()p Fn(\030)35133 18050 y Fq(1)35658 17851 y Fn(;)221 b(\030)36808 18050 y Fq(2)37335 17851 y Fw(\))p Fn(:)3432 20606 y Fw(W)-108 b(e)433 b(can)h(rewrite)g(the)f(\010)15643 20805 y Fs(N)16539 20606 y Fw(,)h(\011)18346 20805 y Fs(N)19242 20606 y Fw(,)g(and)f Fn(K)23768 20124 y Fs(r)23673 20941 y(N)25003 20606 y Fw(b)-36 b(y)433 b(using)h(the)f(functions)g Fn(H)39138 20805 y Fs(N)40468 20606 y Fw(and)43334 20270 y(~)42997 20606 y Fn(H)44075 20805 y Fs(N)44971 20606 y Fw(,)5275 23821 y(\010)6214 24020 y Fs(N)7111 23821 y Fw(\()p Fn(\030)8185 24020 y Fq(1)8711 23821 y Fw(\))1107 b(=)f Fn(e)13045 23273 y Fs(w)24 b(\030)14158 23396 y Fg(1)14674 22347 y Fj(\020)15689 22013 y(Z)16427 25020 y Fk(R)17214 24715 y Fg(2)17214 25299 y(+)18162 23821 y Fw(d)p Fn(x)p Fw(d)p Fn(y)48 b(H)22107 24020 y Fs(N)23002 23821 y Fw(\()p Fn(x)295 b Fw(+)g Fn(y)48 b Fw(\))27374 23485 y(~)27037 23821 y Fn(H)28115 24020 y Fs(N)29011 23821 y Fw(\()p Fn(y)342 b Fw(+)295 b Fn(\030)32368 24020 y Fq(1)32894 23821 y Fw(\))g Fm(\000)35023 22013 y Fj(Z)35761 25020 y Fk(R)36548 25139 y Fg(+)37496 23821 y Fw(d)p Fn(y)39238 23485 y Fw(~)38901 23821 y Fn(H)39979 24020 y Fs(N)40875 23821 y Fw(\()p Fn(y)342 b Fw(+)295 b Fn(\030)44232 24020 y Fq(1)44758 23821 y Fw(\))45264 22347 y Fj(\021)46057 23821 y Fn(;)5203 27642 y Fw(\011)6215 27841 y Fs(N)7111 27642 y Fw(\()p Fn(\030)8185 27841 y Fq(2)8711 27642 y Fw(\))1107 b(=)f Fn(e)13045 27094 y Fi(\000)p Fs(w)24 b(\030)14890 27217 y Fg(2)15406 26168 y Fj(\020)16199 27642 y Fw(1)296 b Fm(\000)18473 25834 y Fj(Z)19211 28841 y Fk(R)19998 28960 y Fg(+)20946 27642 y Fw(d)p Fn(y)48 b(H)23429 27841 y Fs(N)24324 27642 y Fw(\()p Fn(y)343 b Fw(+)295 b Fn(\030)27682 27841 y Fq(2)28208 27642 y Fw(\))28714 26168 y Fj(\021)29507 27642 y Fn(;)13448 b Fw(\(6.29\))3432 31304 y Fn(K)4634 30755 y Fs(r)4539 31632 y(N)5435 31304 y Fw(\()p Fn(\030)6509 31503 y Fq(1)7034 31304 y Fn(;)221 b(\030)8184 31503 y Fq(2)8711 31304 y Fw(\))1107 b(=)f Fn(e)13045 30755 y Fi(\000)p Fs(w)24 b(\030)14890 30878 y Fg(1)15406 31304 y Fn(e)16009 30755 y Fs(w)g(\030)17122 30878 y Fg(2)17860 29496 y Fj(Z)18598 32503 y Fk(R)19385 32622 y Fg(+)20332 31304 y Fw(d)p Fn(xH)22872 31503 y Fs(N)23768 31304 y Fw(\()p Fn(x)295 b Fw(+)g Fn(\030)27183 31503 y Fq(1)27709 31304 y Fw(\))28552 30968 y(~)28215 31304 y Fn(H)29293 31503 y Fs(N)30189 31304 y Fw(\()p Fn(x)g Fw(+)g Fn(\030)33604 31503 y Fq(2)34130 31304 y Fw(\))p Fn(:)5383 34709 y Fw(W)-108 b(e)411 b(w)-36 b(an)g(t)411 b(to)g(a)-36 b(v)g(oid)412 b(to)f(write)h(alw)-36 b(a)g(ys)412 b(the)f(pro)72 b(jection)412 b Fn(P)33954 34908 y Fq(0)34480 34709 y Fw(.)571 b(Therefore)411 b(from)h(no)-36 b(w)3432 36314 y(on)422 b Fm(h\001)p Fn(;)221 b Fm(\001i)423 b Fw(refers)g(to)g(the)f(scalar)h(pro)36 b(duct)422 b(in)h Fn(L)26253 35832 y Fq(2)26778 36314 y Fw(\()p Fl(R)28364 36513 y Fq(+)29155 36314 y Fn(;)221 b Fw(d)p Fn(x)p Fw(\),)425 b Fm(k)273 b(\001)g(k)423 b Fw(is)g(the)f(corresp)36 b(onding)3432 37919 y(norm,)608 b(and)572 b(the)h(in)-36 b(tegral)574 b(op)36 b(erators)573 b(act)g(in)h Fn(L)28026 37437 y Fq(2)28551 37919 y Fw(\()p Fl(R)30137 38118 y Fq(+)30928 37919 y Fn(;)221 b Fw(d)p Fn(x)p Fw(\).)997 b(First)573 b(let)g(us)g(consider)3432 39524 y Fn(w)447 b(>)412 b Fw(0.)654 b(Let)459 b(us)f(denote)g Fn(A)412 b Fw(=)g(\()p Fl(1)315 b Fm(\000)e Fn(K)23022 39747 y Fq(Ai)p Fs(;w)24948 39495 y Fg(2)25409 39747 y Fq(+)p Fs(s)26632 39524 y Fw(\))27138 39042 y Fi(\000)p Fq(1)28395 39524 y Fw(.)654 b(Then)459 b(for)g(\014nite)f Fn(N)597 b Fw(w)-36 b(e)460 b(ha)-36 b(v)g(e)459 b(the)3432 41129 y(b)36 b(ound)6022 43696 y Fm(jh)p Fw(\010)7847 43895 y Fs(N)8743 43696 y Fn(;)221 b(A)10300 43895 y Fs(N)11198 43696 y Fw(\011)12210 43895 y Fs(N)13106 43696 y Fm(i)294 b(\000)i(h)p Fw(\010)16702 43895 y Fs(w)24 b(;s)18153 43696 y Fn(;)221 b(A)p Fw(\011)20722 43895 y Fs(w)24 b(;s)22173 43696 y Fm(ij)20220 b Fw(\(6.30\))6022 45633 y Fm(\024)369 b(k)p Fw(\010)9027 45832 y Fs(N)9924 45633 y Fm(k)221 b(k)p Fn(A)12448 45832 y Fs(N)13641 45633 y Fm(\000)295 b Fn(A)p Fm(k)221 b(k)p Fw(\011)18505 45832 y Fs(N)19402 45633 y Fm(k)295 b Fw(+)g Fm(k)p Fw(\010)23271 45832 y Fs(N)24463 45633 y Fm(\000)h Fw(\010)p Fm(k)221 b(k)p Fn(A)p Fm(k)g(k)p Fw(\011)31816 45832 y Fs(N)32714 45633 y Fm(k)295 b Fw(+)g Fm(k)p Fw(\010)p Fm(k)221 b(k)p Fn(A)p Fm(k)g(k)p Fw(\011)41668 45832 y Fs(N)42861 45633 y Fm(\000)295 b Fw(\011)p Fm(k)p Fn(:)3432 48200 y Fw(In)578 b(Lemma)h(6.4)g(w)-36 b(e)579 b(will)h(pro)-36 b(v)g(e)579 b(that)f(\010)24444 48399 y Fs(N)25920 48200 y Fw(con)-36 b(v)g(erges)579 b(to)g(\010)34563 48399 y Fs(w)24 b(;s)36014 48200 y Fw(,)616 b(\011)38003 48399 y Fs(N)39478 48200 y Fw(con)-36 b(v)g(erges)579 b(to)3432 49805 y(\011)4444 50004 y Fs(w)24 b(;s)5894 49805 y Fw(,)582 b(and)552 b Fn(A)10460 50004 y Fs(N)11909 49805 y Fw(con)-36 b(v)g(erges)553 b(to)f(\()p Fl(1)379 b Fm(\000)d Fn(K)23734 50027 y Fq(Ai)p Fs(;w)25660 49775 y Fg(2)26122 50027 y Fq(+)p Fs(s)27344 49805 y Fw(\))27850 49323 y Fi(\000)p Fq(1)29660 49805 y Fw(in)552 b(op)36 b(erator)552 b(norm)g(\(in)g Fn(L)43195 49323 y Fq(2)43721 49805 y Fw(\()p Fl(R)45307 50004 y Fq(+)46097 49805 y Fw(\))3432 51410 y(according)434 b(to)f(our)h(con) -36 b(v)g(en)g(tion\).)578 b(This)433 b(implies)11696 53976 y(lim)11238 54803 y Fs(N)94 b Fi(!1)13961 53976 y Fm(h)p Fw(\010)15417 54175 y Fs(N)16314 53976 y Fn(;)221 b(A)17871 54175 y Fs(N)18768 53976 y Fw(\011)19780 54175 y Fs(N)20676 53976 y Fm(i)368 b Fw(=)h Fm(h)p Fw(\010)24398 54175 y Fs(w)24 b(;s)25849 53976 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)30444 54199 y Fq(Ai)p Fs(;w)32370 53947 y Fg(2)32832 54199 y Fq(+)p Fs(s)34054 53976 y Fw(\))34560 53428 y Fi(\000)p Fq(1)35817 53976 y Fw(\011)36829 54175 y Fs(w)24 b(;s)38280 53976 y Fm(i)4482 b Fw(\(6.31\))3432 57004 y(whic)-36 b(h)433 b(is)h(precisely)g(the)f(last)h(term)f(in)h (\(6.10\).)5383 58609 y(F)-108 b(or)402 b(the)g(case)h Fn(w)i Fw(=)368 b(0)403 b(w)-36 b(e)404 b(ha)-36 b(v)g(e)402 b(to)h(mo)36 b(dify)404 b(sligh)-36 b(tly)404 b(the)e(argumen)-36 b(t.)567 b(In)403 b(this)f(case)3432 60214 y(\011)4444 60413 y Fs(N)5849 60214 y Fw(is)510 b(not)g(in)f Fn(L)12100 59732 y Fq(2)12626 60214 y Fw(\()p Fl(R)14212 60413 y Fq(+)15002 60214 y Fw(\),)529 b(but)508 b(\011)19870 60413 y Fs(N)20767 60214 y Fw(\()p Fn(\030)64 b Fw(\))346 b Fm(\000)i Fw(1)498 b(=)g Fm(\000)28050 59144 y Fj(R)28678 60675 y Fk(R)29465 60794 y Fg(+)30413 60214 y Fw(d)p Fn(y)48 b(H)32896 60413 y Fs(N)33791 60214 y Fw(\()p Fn(y)395 b Fw(+)347 b Fn(\030)64 b Fw(\))498 b Fm(2)g Fn(L)40590 59732 y Fq(2)41116 60214 y Fw(\()p Fl(R)42702 60413 y Fq(+)43492 60214 y Fw(\))510 b(and)3432 62068 y(con)-36 b(v)g(erges)434 b(to)f Fm(\000)12099 60997 y Fj(R)12727 62529 y Fk(R)13514 62648 y Fg(+)14462 62068 y Fw(d)p Fn(x')16776 62267 y Fq(0)p Fs(;s)17998 62068 y Fw(\()p Fn(x)295 b Fw(+)g Fn(\030)64 b Fw(\))433 b(in)h(the)f Fn(N)507 b Fm(!)369 b(1)434 b Fw(limit.)579 b(W)-108 b(e)434 b(write)11300 64809 y Fm(h)p Fw(\010)12756 65008 y Fs(N)13652 64809 y Fn(;)221 b(A)15209 65008 y Fs(N)16107 64809 y Fw(\011)17119 65008 y Fs(N)18015 64809 y Fm(i)368 b Fw(=)h Fm(h)p Fw(\010)21737 65008 y Fs(N)22633 64809 y Fn(;)221 b(A)24190 65008 y Fs(N)25088 64809 y Fw(\(\011)26606 65008 y Fs(N)27797 64809 y Fm(\000)295 b Fw(1\))p Fm(i)g Fw(+)g Fm(h)p Fw(\010)33856 65008 y Fs(N)34752 64809 y Fn(;)221 b(A)36309 65008 y Fs(N)37206 64809 y Fw(1)p Fm(i)p Fn(:)4545 b Fw(\(6.32\))3432 67375 y(Here)508 b(1)h(denotes)f(the)g(constan)-36 b(t)508 b(function)g Fn( )48 b Fw(\()p Fn(x)p Fw(\))495 b(=)i(1.)803 b(By)509 b(the)f(same)h(argumen)-36 b(t)508 b(as)3432 68980 y(for)434 b Fn(w)404 b(>)369 b Fw(0,)434 b(the)f(\014rst)g(term)g(in)g(\(6.32\))i (con)-36 b(v)g(erges)434 b(as)6568 71547 y(lim)6110 72373 y Fs(N)94 b Fi(!1)8833 71547 y Fm(h)p Fw(\010)10289 71746 y Fs(N)11185 71547 y Fn(;)221 b(A)12742 71746 y Fs(N)13640 71547 y Fw(\(\011)15158 71746 y Fs(N)16349 71547 y Fm(\000)295 b Fw(1\))p Fm(i)369 b Fw(=)f Fm(h)p Fw(\010)22555 71746 y Fs(w)24 b(;s)24006 71547 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)d Fn(K)28602 71769 y Fq(Ai)o Fs(;w)30527 71517 y Fg(2)30989 71769 y Fq(+)p Fs(s)32211 71547 y Fw(\))32717 70998 y Fi(\000)p Fq(1)33975 71547 y Fw(\(\011)35493 71746 y Fs(w)24 b(;s)37238 71547 y Fm(\000)296 b Fw(1\))p Fm(i)p Fn(:)2678 b Fw(\(6.33\))24367 75163 y(25)p eop end end %%Page: 26 26 TeXDict begin HPSdict begin 26 25 bop 3432 7638 a Fw(The)433 b(second)g(term)h(in)f(\(6.32\))h(can)g(b)36 b(e)433 b(rewritten)h(as)13125 10948 y Fm(h)p Fw(\010)14581 11147 y Fs(N)15477 10948 y Fn(;)221 b(A)17034 11147 y Fs(N)17931 10948 y Fw(1)p Fm(i)369 b Fw(=)20848 9140 y Fj(Z)21586 12148 y Fk(R)22373 11843 y Fg(2)22373 12426 y(+)23321 10948 y Fw(d)p Fn(\030)24612 11147 y Fq(1)25137 10948 y Fw(d)p Fn(\030)26428 11147 y Fq(2)26953 10948 y Fw(\010)27892 11147 y Fs(N)28789 10948 y Fw(\()p Fn(\030)29863 11147 y Fq(1)30389 10948 y Fw(\))p Fn(A)31870 11147 y Fs(N)32766 10948 y Fw(\()p Fn(\030)33840 11147 y Fq(1)34366 10948 y Fn(;)221 b(\030)35516 11147 y Fq(2)36043 10948 y Fw(\))p Fn(:)6369 b Fw(\(6.34\))3432 14611 y(In)433 b(Lemma)g(6.5)i(w)-36 b(e)434 b(will)h(pro)-36 b(v)g(e)433 b(the)g(con)-36 b(v)g(ergence)13051 17276 y(lim)12592 18102 y Fs(N)94 b Fi(!1)15315 17276 y Fm(h)p Fw(\010)16771 17475 y Fs(N)17668 17276 y Fn(;)221 b(A)19225 17475 y Fs(N)20122 17276 y Fw(1)p Fm(i)369 b Fw(=)g Fm(h)p Fw(\010)24495 17475 y Fs(w)24 b(;s)25945 17276 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)d Fn(K)30541 17498 y Fq(Ai)p Fs(;w)32467 17246 y Fg(2)32928 17498 y Fq(+)p Fs(s)34150 17276 y Fw(\))34656 16727 y Fi(\000)p Fq(1)35914 17276 y Fw(1)p Fm(i)p Fn(:)5837 b Fw(\(6.35\))3432 21066 y Fo(Thir)-66 b(d)409 b(term)h(of)g(\(5.22\))i (for)e Fn(w)404 b Fm(\024)369 b Fw(0)p Fo(.)559 b Fw(The)374 b(computations)g(for)h(this)f(case)h(are)g(as)g(b)36 b(efore)3432 22671 y(and)433 b(the)g(third)f(term)h(of)i(\(5.22\))f (con)-36 b(v)g(erges)434 b(to)16178 25335 y Fm(h)p Fw(\011)17707 25534 y Fi(\000)p Fs(w)24 b(;s)19889 25335 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)24484 25558 y Fq(Ai)p Fs(;w)26410 25306 y Fg(2)26871 25558 y Fq(+)p Fs(s)28094 25335 y Fw(\))28600 24787 y Fi(\000)p Fq(1)29857 25335 y Fw(\010)30796 25534 y Fi(\000)p Fs(w)24 b(;s)32979 25335 y Fm(i)p Fn(:)9422 b Fw(\(6.36\))3432 28000 y(Since)433 b Fn(K)7935 28223 y Fq(Ai)o Fs(;w)9860 27970 y Fg(2)10322 28223 y Fq(+)p Fs(s)11978 28000 y Fw(is)h(symmetric,)g(this)f (concludes)h(the)f(pro)36 b(of)434 b(of)g(Theorem)g(6.2.)p 45663 30269 45 878 v 45708 29436 781 45 v 45708 30269 V 46487 30269 45 878 v 3432 32751 a Fh(Lemma)499 b(6.4.)554 b Fo(L)-66 b(et)463 b Fn(w)405 b Fm(\025)369 b Fw(0)p Fo(.)598 b(Then)16625 35415 y Fw(lim)16167 36242 y Fs(N)94 b Fi(!1)19111 35415 y Fm(k)p Fw(\010)20714 35614 y Fs(N)21906 35415 y Fm(\000)296 b Fw(\010)24174 35614 y Fs(w)24 b(;s)25625 35415 y Fm(k)369 b Fw(=)g(0)p Fn(;)16625 37814 y Fw(lim)16167 38640 y Fs(N)94 b Fi(!1)19111 37814 y Fm(k)p Fw(\011)20787 38013 y Fs(N)21979 37814 y Fm(\000)295 b Fw(\011)24319 38013 y Fs(w)24 b(;s)25770 37814 y Fm(k)369 b Fw(=)f(0)p Fn(;)14085 b Fw(\(6.37\))16625 40335 y(lim)16167 41162 y Fs(N)94 b Fi(!1)19111 40335 y Fm(k)p Fn(A)20750 40534 y Fs(N)21942 40335 y Fm(\000)295 b Fw(\()p Fl(1)k Fm(\000)c Fn(K)27283 40558 y Fq(Ai)p Fs(;w)29209 40306 y Fg(2)29671 40558 y Fq(+)p Fs(s)30893 40335 y Fw(\))31399 39787 y Fi(\000)p Fq(1)32656 40335 y Fm(k)369 b Fw(=)g(0)p Fn(;)3432 43461 y Fo(wher)-66 b(e)409 b(the)g(functions)f Fw(\010)15678 43660 y Fs(N)16575 43461 y Fn(;)221 b Fw(\011)18169 43660 y Fs(N)19066 43461 y Fo(,)420 b(and)410 b(the)f(inte)-66 b(gr)g(al)407 b(kernel)i Fn(A)33823 43660 y Fs(N)35129 43461 y Fo(ar)-66 b(e)409 b(de\014ne)-66 b(d)409 b(in)f(\(6.28\))3432 45066 y(and)464 b(\(6.29\).)3432 47548 y(Pr)-66 b(o)g(of.)649 b(Conver)-66 b(genc)g(e)412 b(of)j Fw(\010)17393 47747 y Fs(N)18290 47548 y Fo(.)560 b Fw(Let)379 b(us)h(consider,)390 b(for)381 b(an)-36 b(y)380 b(\014xed)f Fn(\030)434 b Fm(\025)369 b Fw(0,)391 b(the)379 b(function)3432 49153 y(\010)4371 49352 y Fs(N)5267 49153 y Fw(\()p Fn(\030)64 b Fw(\))434 b(de\014ned)e(in)h(\(6.29\).)580 b(W)-108 b(e)433 b(\014rst)g(sho)-36 b(w)434 b(that)19372 51817 y(lim)18913 52644 y Fs(N)94 b Fi(!1)21858 51817 y Fw(\010)22797 52016 y Fs(N)23694 51817 y Fw(\()p Fn(\030)64 b Fw(\))368 b(=)h(\010)28026 52016 y Fs(w)24 b(;s)29477 51817 y Fw(\()p Fn(\030)64 b Fw(\))12158 b(\(6.38\))3432 55164 y(p)36 b(oin)-36 b(t)g(wise.)1128 b(Using)617 b(the)g(exp)36 b(onen)-36 b(tial)617 b(deca)-36 b(y)617 b(\(6.16\))h(of)g Fn(H)34138 55363 y Fs(N)35651 55164 y Fw(and)38701 54828 y(~)38363 55164 y Fn(H)39441 55363 y Fs(N)40338 55164 y Fw(,)663 b(w)-36 b(e)617 b(apply)3432 56769 y(dominated)491 b(con)-36 b(v)g(ergence)492 b(and)g(exc)-36 b(hange)492 b(the)g(in)-36 b(tegrals)492 b(with)h(the)e Fn(N)607 b Fm(!)469 b(1)492 b Fw(limit.)3432 58374 y(Then)433 b(using)g(the)g(p)36 b(oin)-36 b(t)g(wise)434 b(limit)g(of)g Fn(H)23889 58573 y Fs(N)24786 58374 y Fw(,)f(see)h(\(6.17\),)h(one)f (obtains)3890 61685 y(lim)3432 62511 y Fs(N)94 b Fi(!1)6376 61685 y Fw(\010)7315 61884 y Fs(N)8212 61685 y Fw(\()p Fn(\030)64 b Fw(\))369 b(=)f Fn(e)12208 61136 y Fs(w)24 b(\030)13632 59877 y Fj(Z)14370 62884 y Fk(R)15157 62579 y Fg(2)15157 63163 y(+)16105 61685 y Fw(d)p Fn(y)48 b Fw(d)p Fn(x')19824 61884 y Fi(\000)p Fs(w)24 b(;s)22005 61685 y Fw(\()p Fn(y)278 b Fw(+)230 b Fn(\030)64 b Fw(\))p Fn(')26655 61884 y Fs(w)24 b(;s)28106 61685 y Fw(\()p Fn(x)231 b Fw(+)f Fn(y)48 b Fw(\))230 b Fm(\000)g Fn(e)34108 61136 y Fs(w)24 b(\030)35532 59877 y Fj(Z)36270 62884 y Fk(R)37057 63003 y Fg(+)38004 61685 y Fw(d)p Fn(y)48 b(')40261 61884 y Fi(\000)p Fs(w)24 b(;s)42444 61685 y Fw(\()p Fn(y)278 b Fw(+)230 b Fn(\030)64 b Fw(\))p Fn(;)43279 64362 y Fw(\(6.39\))3432 65967 y(whic)-36 b(h)611 b(is)h(precisely)g(\010)15252 66166 y Fs(w)24 b(;s)16703 65967 y Fw(\()p Fn(\030)64 b Fw(\).)1112 b(By)612 b(the)f(exp)36 b(onen)-36 b(tial)612 b(deca)-36 b(y)612 b(\(6.16\))g(with)f Fn(\014)746 b(>)672 b(w)3432 67572 y Fw(it)563 b(follo)-36 b(ws)565 b(that)e(\010)13149 67771 y Fs(N)14635 67572 y Fm(2)589 b Fn(L)16995 67090 y Fq(2)17521 67572 y Fw(\()p Fl(R)19107 67771 y Fq(+)19897 67572 y Fw(\).)967 b(Moreo)-36 b(v)g(er,)596 b(\010)28901 67771 y Fs(N)30361 67572 y Fw(is)564 b(uniformly)f(b)36 b(ounded)562 b(b)-36 b(y)563 b(an)3432 69177 y(in)-36 b(tegrable)433 b(function)g(since,)h(for)h(all)f Fn(\014)443 b(>)369 b(w)36 b Fw(,)16063 71842 y Fm(j)p Fw(\010)17371 72041 y Fs(N)18268 71842 y Fw(\()p Fn(\030)64 b Fw(\))p Fm(j)369 b(\024)g Fn(C)23078 71293 y Fq(2)22983 72170 y Fs(\014)23612 71842 y Fn(e)24215 71293 y Fi(\000)p Fs(\014)46 b(x)26105 71842 y Fn(e)26708 71293 y Fi(\000)p Fq(2)p Fs(\014)g(y)29036 71842 y Fn(e)29639 71293 y Fi(\000)p Fq(\()p Fs(\014)g Fi(\000)p Fs(w)24 b Fq(\))p Fs(\030)33610 71842 y Fn(:)9308 b Fw(\(6.40\))24367 75163 y(26)p eop end end %%Page: 27 27 TeXDict begin HPSdict begin 27 26 bop 3432 7638 a Fw(Therefore,)514 b(b)-36 b(y)498 b(dominated)g(con)-36 b(v)g(ergence)498 b(and)g(b)-36 b(y)497 b(p)36 b(oin)-36 b(t)g(wise)499 b(con)-36 b(v)g(ergence)498 b(\(6.39\),)3432 9243 y(w)-36 b(e)433 b(obtain)7537 12199 y(lim)7079 13025 y Fs(N)94 b Fi(!1)10023 12199 y Fm(k)p Fw(\010)11626 12398 y Fs(N)12818 12199 y Fm(\000)296 b Fw(\010)15086 12398 y Fs(w)24 b(;s)16537 12199 y Fm(k)17201 11650 y Fq(2)18096 12199 y Fw(=)19476 10391 y Fj(Z)20214 13398 y Fk(R)21001 13517 y Fg(+)21949 12199 y Fw(d)p Fn(\030)743 b Fw(lim)23525 13025 y Fs(N)94 b Fi(!1)26470 12199 y Fm(j)p Fw(\010)27778 12398 y Fs(N)28674 12199 y Fw(\()p Fn(\030)64 b Fw(\))296 b Fm(\000)f Fw(\010)32881 12398 y Fs(w)24 b(;s)34332 12199 y Fw(\()p Fn(\030)64 b Fw(\))p Fm(j)36345 11650 y Fq(2)37240 12199 y Fw(=)368 b(0)p Fn(:)3648 b Fw(\(6.41\))3432 16417 y Fo(Conver)-66 b(genc)g(e)495 b(of)j Fw(\011)13560 16616 y Fs(N)14456 16417 y Fo(.)687 b Fw(Let)469 b(us)h(consider,)479 b(for)471 b Fn(\030)495 b Fm(\025)431 b Fw(0,)479 b(the)470 b(function)g(\011) 39547 16616 y Fs(N)40443 16417 y Fw(\()p Fn(\030)64 b Fw(\))470 b(de\014ned)3432 18022 y(in)433 b(\(6.29\).)579 b(W)-108 b(e)434 b(\014rst)f(sho)-36 b(w)433 b(that)19299 20664 y(lim)18841 21491 y Fs(N)94 b Fi(!1)21785 20664 y Fw(\011)22797 20863 y Fs(N)23694 20664 y Fw(\()p Fn(\030)64 b Fw(\))368 b(=)h(\011)28099 20863 y Fs(w)24 b(;s)29550 20664 y Fw(\()p Fn(\030)64 b Fw(\))12085 b(\(6.42\))3432 23767 y(p)36 b(oin)-36 b(t)g(wise.)574 b(As)421 b(b)36 b(efore,)424 b(\(6.16\))e(allo)-36 b(ws)423 b(us)d(to)h(exc)-36 b(hange)422 b(the)e(limit)i(and)e(the)h(in)-36 b(tegral)3432 25373 y(with)433 b(the)g(result)12882 28359 y(lim)12424 29185 y Fs(N)94 b Fi(!1)15369 28359 y Fw(\011)16381 28558 y Fs(N)17277 28359 y Fw(\()p Fn(\030)64 b Fw(\))369 b(=)f Fn(e)21273 27810 y Fi(\000)p Fs(w)24 b(\030)23207 26884 y Fj(\020)24001 28359 y Fw(1)295 b Fm(\000)26275 26551 y Fj(Z)27603 26901 y Fi(1)27013 29558 y Fq(0)28821 28359 y Fw(d)p Fn(y)48 b(')31078 28558 y Fs(w)24 b(;s)32528 28359 y Fw(\()p Fn(y)343 b Fw(+)295 b Fn(\030)64 b Fw(\))36456 26884 y Fj(\021)37249 28359 y Fn(:)5669 b Fw(\(6.43\))3432 31758 y(Then)632 b(b)-36 b(y)632 b(\(6.16\),)684 b(with)632 b Fn(\014)782 b(>)707 b(w)36 b Fw(,)682 b(\011)22817 31957 y Fs(N)24421 31758 y Fm(2)707 b Fn(L)26899 31276 y Fq(2)27425 31758 y Fw(\()p Fl(R)29011 31957 y Fq(+)29801 31758 y Fw(\))632 b(and)g(\011)34679 31957 y Fs(N)36208 31758 y Fw(is)h(b)36 b(ounded)631 b(b)-36 b(y)632 b(an)3432 33363 y(in)-36 b(tegrable)433 b(function.)578 b(Therefore)7392 36577 y(lim)6934 37404 y Fs(N)94 b Fi(!1)9879 36577 y Fm(k)p Fw(\011)11555 36776 y Fs(N)12746 36577 y Fm(\000)295 b Fw(\011)15086 36776 y Fs(w)24 b(;s)16537 36577 y Fm(k)17201 36029 y Fq(2)18096 36577 y Fw(=)19476 34769 y Fj(Z)20214 37776 y Fk(R)21001 37895 y Fg(+)21949 36577 y Fw(d)p Fn(\030)743 b Fw(lim)23525 37404 y Fs(N)94 b Fi(!1)26470 36577 y Fm(j)p Fw(\011)27851 36776 y Fs(N)28747 36577 y Fw(\()p Fn(\030)64 b Fw(\))295 b Fm(\000)g Fw(\011)33026 36776 y Fs(w)24 b(;s)34477 36577 y Fw(\()p Fn(\030)64 b Fw(\))p Fm(j)36490 36029 y Fq(2)37384 36577 y Fw(=)369 b(0)p Fn(:)3503 b Fw(\(6.44\))3432 40852 y Fo(Conver)-66 b(genc)g(e)461 b(of)k Fn(A)13456 41051 y Fs(N)14353 40852 y Fo(.)577 b Fw(Denote)434 b Fn(q)416 b Fw(=)369 b Fn(w)23131 40370 y Fq(2)23951 40852 y Fw(+)295 b Fn(s)p Fw(.)578 b(W)-108 b(e)434 b(w)-36 b(an)g(t)433 b(to)h(sho)-36 b(w)434 b(that)14236 43494 y(lim)13778 44321 y Fs(N)94 b Fi(!1)16722 43494 y Fm(k)p Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)21494 42946 y Fs(r)21399 43823 y(N)22295 43494 y Fw(\))22801 42946 y Fi(\000)p Fq(1)24354 43494 y Fm(\000)g Fw(\()p Fl(1)k Fm(\000)c Fn(K)29695 43693 y Fq(Ai)p Fs(;q)31429 43494 y Fw(\))31935 42946 y Fi(\000)p Fq(1)33193 43494 y Fm(k)369 b Fw(=)f(0)7023 b(\(6.45\))3432 46524 y(in)433 b(op)36 b(erator)434 b(norm.)578 b(Assume)433 b(that)g(w)-36 b(e)434 b(can)g(sho)-36 b(w:)3432 48793 y(1\))433 b(lim)6828 48992 y Fs(N)94 b Fi(!1)9827 48793 y Fm(k)p Fn(K)11693 48311 y Fs(r)11598 49128 y(N)12790 48793 y Fm(\000)295 b Fn(K)15225 48992 y Fq(Ai)p Fs(;q)16960 48793 y Fm(k)369 b Fw(=)f(0,)3432 51062 y(2\))433 b Fm(k)p Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)9698 51261 y Fq(Ai)p Fs(;q)11432 51062 y Fw(\))11938 50580 y Fi(\000)p Fq(1)13196 51062 y Fm(k)369 b Fn(<)g Fm(1)p Fw(.)3432 53331 y(Then)433 b(with)g(the)g(notation)h Fn(K)18340 53530 y Fs(")19198 53331 y Fw(=)369 b Fn(K)21686 53530 y Fq(Ai)p Fs(;q)23716 53331 y Fm(\000)295 b Fn(K)26246 52849 y Fs(r)26151 53667 y(N)27481 53331 y Fw(w)-36 b(e)433 b(ha)-36 b(v)g(e)12639 55973 y Fm(k)p Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)17411 55425 y Fs(r)17316 56302 y(N)18212 55973 y Fw(\))18718 55425 y Fi(\000)p Fq(1)20271 55973 y Fm(\000)g Fw(\()p Fl(1)j Fm(\000)e Fn(K)25612 56172 y Fq(Ai)26633 55973 y Fw(\))27139 55425 y Fi(\000)p Fq(1)28396 55973 y Fm(k)12639 57910 y Fw(=)369 b Fm(k)p Fw([\()p Fl(1)298 b Fw(+)d(\()p Fl(1)k Fm(\000)c Fn(K)21942 58109 y Fq(Ai)p Fs(;q)23676 57910 y Fw(\))24182 57362 y Fi(\000)p Fq(1)25440 57910 y Fn(K)26547 58109 y Fs(")27036 57910 y Fw(\))27542 57362 y Fi(\000)p Fq(1)29095 57910 y Fm(\000)g Fw(1]\()p Fl(1)k Fm(\000)d Fn(K)35448 58109 y Fq(Ai)o Fs(;q)37182 57910 y Fw(\))37688 57362 y Fi(\000)p Fq(1)38945 57910 y Fm(k)12639 60119 y(\024)369 b(k)p Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)18718 60318 y Fq(Ai)p Fs(;q)20452 60119 y Fw(\))20958 59571 y Fi(\000)p Fq(1)22216 60119 y Fm(k)23101 58857 y Fj(X)23174 61663 y Fs(n)p Fi(\025)p Fq(1)25242 60119 y Fm(k)p Fw(\()p Fl(1)j Fm(\000)d Fn(K)29918 60318 y Fq(Ai)p Fs(;q)31653 60119 y Fw(\))32159 59571 y Fi(\000)p Fq(1)33416 60119 y Fm(k)34080 59571 y Fs(n)34707 60119 y Fm(k)p Fn(K)36478 60318 y Fs(")36967 60119 y Fm(k)37631 59571 y Fs(n)43279 60119 y Fw(\(6.46\))3432 64089 y(whic)-36 b(h)453 b(con)-36 b(v)g(erges)454 b(to)g(0)g(as)g Fn(N)542 b Fm(!)404 b(1)p Fw(.)639 b(So)454 b(w)-36 b(e)454 b(ha)-36 b(v)g(e)454 b(to)g(establish)f(prop)36 b(erties)454 b(1\))g(and)3432 65694 y(2\).)541 b(Let)322 b(us)h(start)f(with)h(1\).)542 b(F)-108 b(or)322 b(\014xed)g Fn(x;)221 b(y)418 b Fm(\025)369 b Fw(0,)346 b(using)322 b(\(6.16\))i(with)e Fn(\014)443 b(>)369 b(w)36 b Fw(,)345 b Fn(K)42787 65212 y Fs(r)42692 66030 y(N)43588 65694 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))3432 67300 y(is)400 b(uniformly)h(b)36 b(ounded)399 b(b)-36 b(y)400 b(a)h(function)f(whic)-36 b(h)400 b(is)g(in)-36 b(tegrable)400 b(in)h Fl(R)37288 66818 y Fq(2)37288 67628 y(+)38078 67300 y Fw(.)567 b(Therefore,)408 b(b)-36 b(y)3432 68905 y(dominated)433 b(con)-36 b(v)g(ergence)433 b(and)g(the)h(p)36 b(oin)-36 b(t)g(wise)433 b(limit)7663 71547 y(lim)7205 72373 y Fs(N)94 b Fi(!1)10150 71547 y Fn(H)11228 71746 y Fs(N)12124 71547 y Fw(\()p Fn(x)295 b Fw(+)g Fn(z)59 b Fw(\))16477 71211 y(~)16140 71547 y Fn(H)17218 71746 y Fs(N)18114 71547 y Fw(\()p Fn(y)343 b Fw(+)295 b Fn(z)59 b Fw(\))369 b(=)f(Ai)q(\()p Fn(q)342 b Fw(+)295 b Fn(x)g Fw(+)g Fn(z)59 b Fw(\))221 b(Ai)q(\()p Fn(q)342 b Fw(+)295 b Fn(y)343 b Fw(+)295 b Fn(z)59 b Fw(\))p Fn(;)3774 b Fw(\(6.47\))24367 75163 y(27)p eop end end %%Page: 28 28 TeXDict begin HPSdict begin 28 27 bop 3432 7638 a Fw(one)433 b(obtains)6267 10571 y(lim)5809 11398 y Fs(N)94 b Fi(!1)8753 10571 y Fm(j)p Fn(K)10324 10023 y Fs(r)10229 10900 y(N)11125 10571 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))296 b Fm(\000)f Fn(K)16871 10770 y Fq(Ai)p Fs(;q)18605 10571 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))p Fm(j)21290 b Fw(\(6.48\))5809 13812 y Fm(\024)7211 12004 y Fj(Z)8539 12355 y Fi(1)7949 15012 y Fq(0)9757 13812 y Fw(d)p Fn(z)59 b Fw([)458 b(lim)11504 14639 y Fs(N)94 b Fi(!1)14449 13812 y Fn(H)15527 14011 y Fs(N)16423 13812 y Fw(\()p Fn(x)295 b Fw(+)g Fn(z)59 b Fw(\))20776 13477 y(~)20439 13812 y Fn(H)21517 14011 y Fs(N)22413 13812 y Fw(\()p Fn(y)343 b Fw(+)295 b Fn(z)59 b Fw(\))295 b Fm(\000)g Fw(Ai)q(\()p Fn(q)342 b Fw(+)295 b Fn(x)g Fw(+)g Fn(z)59 b Fw(\))221 b(Ai)q(\()p Fn(q)342 b Fw(+)295 b Fn(y)343 b Fw(+)295 b Fn(z)59 b Fw(\)])370 b(=)e(0)p Fn(:)3432 17373 y Fw(Moreo)-36 b(v)g(er,)406 b(it)399 b(is)g(easy)h(to)f(see)g(that)f Fn(K)22447 16891 y Fs(r)22352 17709 y(N)23647 17373 y Fw(is)h(a)g(Hilb)36 b(ert-Sc)-36 b(hmidt)397 b(op)36 b(erator)400 b(with)e(norm)3432 18978 y(uniformly)434 b(b)36 b(ounded)432 b(in)h Fn(N)139 b Fw(,)433 b(and)h(so)f(is)h Fn(K)24701 19177 y Fq(Ai)p Fs(;q)26436 18978 y Fw(.)578 b(Therefore)8021 21912 y(lim)7563 22738 y Fs(N)94 b Fi(!1)10508 21912 y Fm(k)p Fn(K)12374 21363 y Fs(r)12279 22240 y(N)13175 21912 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))295 b Fm(\000)h Fn(K)18921 22111 y Fq(Ai)o Fs(;q)20655 21912 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))p Fm(k)24334 21363 y Fq(2)25229 21912 y Fm(\024)828 b Fw(lim)26631 22738 y Fs(N)94 b Fi(!1)29576 21912 y Fm(k)p Fn(K)31442 21363 y Fs(r)31347 22240 y(N)32243 21912 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))295 b Fm(\000)h Fn(K)37989 22111 y Fq(Ai)o Fs(;q)39723 21912 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))p Fm(k)43402 21363 y Fq(2)43402 22240 y(HS)7563 25122 y Fw(=)8944 23314 y Fj(Z)9682 26321 y Fk(R)10469 26016 y Fg(2)10469 26600 y(+)11416 25122 y Fw(d)p Fn(x)p Fw(d)p Fn(y)727 b Fw(lim)14504 25948 y Fs(N)94 b Fi(!1)17448 25122 y Fm(j)p Fn(K)19019 24573 y Fs(r)18924 25450 y(N)19820 25122 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))296 b Fm(\000)f Fn(K)25566 25321 y Fq(Ai)p Fs(;q)27300 25122 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))p Fm(j)30684 24573 y Fq(2)31579 25122 y Fw(=)369 b(0)p Fn(:)9308 b Fw(\(6.49\))3432 29127 y(Next)396 b(consider)f(p)36 b(oin)-36 b(t)395 b(2\).)566 b Fl(1)218 b Fm(\000)g Fn(K)20390 29326 y Fq(Ai)r Fs(;q)22523 29127 y Fw(is)395 b(in)-36 b(v)g(ertible)396 b(as)g(b)36 b(ounded)394 b(op)36 b(erator,)404 b(since)395 b(for)3432 30732 y(ev)-36 b(ery)474 b Fn(q)483 b Fm(2)436 b Fl(R)p Fw(,)487 b Fm(k)p Fn(K)12985 30931 y Fq(Ai)p Fs(;q)14719 30732 y Fm(k)437 b Fn(<)f Fw(1.)698 b(T)-108 b(o)473 b(establish)g(the)g(claim,)484 b(let)474 b(us)f(denote)f(b)-36 b(y)473 b Fn(K)43302 30931 y Fq(Ai)44797 30732 y Fw(the)3432 32338 y(standard)432 b(Airy)i(op)36 b(erator)434 b(with)g(Airy)g(k)-36 b(ernel)434 b Fn(K)27921 32537 y Fq(Ai)o Fs(;)p Fq(0)29674 32338 y Fw(.)578 b(Then)16391 35271 y Fm(k)p Fn(K)18162 35470 y Fq(Ai)p Fs(;q)19896 35271 y Fm(k)20560 35494 y Fs(L)21200 35241 y Fg(2)21662 35494 y Fq(\()p Fk(R)22815 35613 y Fg(+)23485 35494 y Fq(\))24276 35271 y Fw(=)368 b Fm(k)p Fn(K)27427 35470 y Fq(Ai)28448 35271 y Fm(k)29112 35494 y Fs(L)29752 35241 y Fg(2)30214 35494 y Fq(\([)p Fs(q)32 b(;)p Fi(1)p Fq(\)\))33282 35271 y Fn(:)9636 b Fw(\(6.50\))3432 38204 y Fn(K)4539 38403 y Fq(Ai)6109 38204 y Fw(is)549 b(an)g(op)36 b(erator)550 b(on)f Fn(L)17616 37722 y Fq(2)18142 38204 y Fw(\([)p Fn(q)48 b(;)221 b Fm(1)p Fw(\)\))549 b(whic)-36 b(h)549 b(is)g(Hilb)36 b(ert-Sc)-36 b(hmidt.)924 b(Therefore)550 b(the)3432 39810 y(norm)453 b(of)h Fn(K)9456 40009 y Fq(Ai)p Fs(;q)11644 39810 y Fw(on)f Fn(L)14355 39327 y Fq(2)14881 39810 y Fw(\([)p Fn(q)48 b(;)221 b Fm(1)p Fw(\)\))454 b(equals)g(its)g(largest) g(eigen)-36 b(v)-72 b(alue,)459 b Fn(\025)37125 40009 y Fq(0)37651 39810 y Fw(\()p Fn(q)48 b Fw(\).)637 b(In)453 b([21)q(])h(it)g(is)3432 41415 y(sho)-36 b(wn)440 b(that)f Fn(\025)10967 41614 y Fq(0)11493 41415 y Fw(\()p Fn(q)48 b Fw(\))439 b(is)i(monotonically)g(decreasing)g(in)f Fn(q)487 b Fw(and)440 b(is)h(strictly)g(less)f(than)g(1)3432 43020 y(for)434 b(an)-36 b(y)434 b Fn(q)416 b(>)368 b Fm(\0001)435 b Fw(\(it)e(con)-36 b(v)g(erges)434 b(to)g(1)g(as)g Fn(q)416 b Fm(!)369 b(\0001)p Fw(\).)p 45663 43020 45 878 v 45708 42186 781 45 v 45708 43020 V 46487 43020 45 878 v 3432 45732 a Fh(Lemma)499 b(6.5.)554 b Fo(L)-66 b(et)463 b Fn(w)405 b Fw(=)369 b(0)p Fo(,)464 b(then)14111 48665 y Fw(lim)13653 49492 y Fs(N)94 b Fi(!1)16376 48665 y Fm(h)p Fw(\010)17832 48864 y Fs(N)18728 48665 y Fn(;)221 b(A)20285 48864 y Fs(N)21182 48665 y Fw(1)p Fm(i)369 b Fw(=)g Fm(h)p Fw(\010)25555 48864 y Fq(0)p Fs(;s)26777 48665 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)d Fn(K)31373 48864 y Fq(Ai)o Fs(;s)33090 48665 y Fw(\))33596 48117 y Fi(\000)p Fq(1)34854 48665 y Fw(1)p Fm(i)p Fn(:)6897 b Fw(\(6.51\))3432 51986 y Fo(Pr)-66 b(o)g(of.)649 b Fw(W)-108 b(e)433 b(\014rst)g(rewrite)11734 54920 y Fm(h)p Fw(\010)13190 55119 y Fs(N)14087 54920 y Fn(;)221 b(A)15644 55119 y Fs(N)16541 54920 y Fw(1)p Fm(i)369 b Fw(=)g Fm(h)p Fw(\010)20914 55119 y Fs(N)21810 54920 y Fn(;)221 b Fw(1)p Fm(i)296 b Fw(+)f Fm(h)p Fw(\010)26618 55119 y Fs(N)27514 54920 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)32204 54371 y Fs(r)32109 55248 y(N)33006 54920 y Fw(\))33512 54371 y Fi(\000)p Fq(1)34769 54920 y Fn(K)35971 54371 y Fs(r)35876 55248 y(N)36772 54920 y Fw(1)p Fm(i)p Fn(:)4979 b Fw(\(6.52\))3432 57853 y(The)757 b(\014rst)g(term,)839 b Fm(h)p Fw(\010)14764 58052 y Fs(N)15660 57853 y Fn(;)221 b Fw(1)p Fm(i)921 b Fw(=)20263 56783 y Fj(R)20890 58314 y Fk(R)21677 58433 y Fg(+)22625 57853 y Fw(d)p Fn(x)p Fw(\010)25026 58052 y Fs(N)25922 57853 y Fw(\()p Fn(x)p Fw(\),)839 b(con)-36 b(v)g(erges)758 b(to)36935 56783 y Fj(R)37562 58314 y Fk(R)38349 58433 y Fg(+)39297 57853 y Fw(d)p Fn(x)p Fw(\010)41698 58052 y Fq(0)p Fs(;s)42920 57853 y Fw(\()p Fn(x)p Fw(\))920 b(=)3432 59689 y Fm(h)p Fw(\010)4888 59888 y Fq(0)p Fs(;s)6110 59689 y Fn(;)221 b Fw(1)p Fm(i)578 b Fw(as)f Fn(N)753 b Fm(!)614 b(1)577 b Fw(b)36 b(ecause)578 b(\010)21715 59888 y Fs(N)23225 59689 y Fm(2)614 b Fn(L)25610 59207 y Fq(1)26135 59689 y Fw(\()p Fl(R)27721 59888 y Fq(+)28512 59689 y Fw(\))577 b(with)g(norm)g Fn(L)37126 59207 y Fq(1)37652 59689 y Fw(\()p Fl(R)39238 59888 y Fq(+)40028 59689 y Fw(\))g(uniformly)3432 61466 y(b)36 b(ounded)533 b(in)i Fn(N)139 b Fw(.)883 b(F)-108 b(or)535 b(the)f(second)h(term,)561 b(de\014ne)29516 61131 y Fj(b)29416 61466 y Fw(\010)30355 61665 y Fs(N)31794 61466 y Fw(=)541 b(\()p Fl(1)368 b Fm(\000)c Fn(K)37593 60837 y Fs(r)-26 b(;)p Fi(\003)37498 61827 y Fs(N)38779 61466 y Fw(\))39285 60984 y Fi(\000)p Fq(1)40542 61466 y Fw(\010)41481 61665 y Fs(N)42378 61466 y Fw(.)884 b(F)-108 b(rom)3432 63232 y(Lemma)378 b(6.4)i(it)f(follo)-36 b(ws)381 b(that,)390 b(for)379 b Fn(N)518 b Fw(large)379 b(enough,)30173 62897 y Fj(b)30072 63232 y Fw(\010)31011 63431 y Fs(N)32277 63232 y Fm(2)369 b Fn(L)34417 62750 y Fq(2)34942 63232 y Fw(\()p Fl(R)36528 63431 y Fq(+)37319 63232 y Fw(\))378 b(with)h(uniformly)3432 64838 y(b)36 b(ounded)432 b(norm.)578 b(Then)4567 68085 y Fm(h)p Fw(\010)6023 68284 y Fs(N)6919 68085 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)d Fn(K)11610 67536 y Fs(r)11515 68413 y(N)12411 68085 y Fw(\))12917 67536 y Fi(\000)p Fq(1)14174 68085 y Fn(K)15376 67536 y Fs(r)15281 68413 y(N)16177 68085 y Fw(1)p Fm(i)369 b Fw(=)19094 66277 y Fj(Z)19832 69284 y Fk(R)20619 69403 y Fg(+)21566 68085 y Fw(d)p Fn(x)23249 66277 y Fj(Z)23987 69284 y Fk(R)24774 69403 y Fg(+)25722 68085 y Fw(d)p Fn(y)27227 67749 y Fj(b)27127 68085 y Fw(\010)28066 68284 y Fs(N)28962 68085 y Fw(\()p Fn(y)48 b Fw(\))p Fn(K)31858 67536 y Fs(r)31763 68413 y(N)32659 68085 y Fw(\()p Fn(y)g(;)221 b(x)p Fw(\))369 b Fm(\021)37445 66277 y Fj(Z)38183 69284 y Fk(R)38970 69403 y Fg(+)39918 68085 y Fw(d)p Fn(xD)42460 68284 y Fs(N)43356 68085 y Fw(\()p Fn(x)p Fw(\))p Fn(:)43279 70602 y Fw(\(6.53\))24367 75163 y(28)p eop end end %%Page: 29 29 TeXDict begin HPSdict begin 29 28 bop 3432 7638 a Fn(D)4512 7837 y Fs(N)5408 7638 y Fw(\()p Fn(x)p Fw(\))506 b(is)g(b)36 b(ounded)505 b(b)-36 b(y)506 b(an)g Fn(L)19072 7156 y Fq(1)20104 7638 y Fw(function)g(indep)36 b(enden)-36 b(t)504 b(of)j Fn(N)139 b Fw(.)795 b(In)506 b(fact,)526 b(using)506 b(the)3432 9243 y(represen)-36 b(tation)432 b(\(6.29\))j(of)f Fn(K)18370 8761 y Fs(r)18275 9579 y(N)19171 9243 y Fw(,)g(one)g(writes)6561 12826 y Fm(j)p Fn(D)8010 13025 y Fs(N)8907 12826 y Fw(\()p Fn(x)p Fw(\))p Fm(j)1106 b(\024)14273 11018 y Fj(Z)15011 14025 y Fs(R)15720 14144 y Fg(+)16673 12826 y Fw(d)p Fn(\025)p Fm(j)18861 12490 y Fw(~)18524 12826 y Fn(H)19602 13025 y Fs(N)20498 12826 y Fw(\()p Fn(x)295 b Fw(+)g Fn(\025)p Fw(\))p Fm(j)25200 11018 y Fj(Z)25938 14025 y Fk(R)26725 14144 y Fg(+)27672 12826 y Fw(d)p Fn(y)48 b Fm(j)p Fn(H)30524 13025 y Fs(N)31420 12826 y Fw(\()p Fn(y)343 b Fw(+)294 b Fn(\025)p Fw(\))p Fm(j)221 b(j)36534 12490 y Fj(b)36433 12826 y Fw(\010)37372 13025 y Fs(N)38269 12826 y Fw(\()p Fn(y)48 b Fw(\))p Fm(j)2947 b Fw(\(6.54\))12133 16521 y Fm(\024)14273 14713 y Fj(Z)15011 17720 y Fs(R)15720 17839 y Fg(+)16673 16521 y Fw(d)p Fn(\025)p Fm(j)18861 16185 y Fw(~)18524 16521 y Fn(H)19602 16720 y Fs(N)20498 16521 y Fw(\()p Fn(x)295 b Fw(+)g Fn(\025)p Fw(\))p Fm(j)24979 15046 y Fj(\020)25993 14713 y(Z)26731 17720 y Fk(R)27518 17839 y Fg(+)28465 16521 y Fw(d)p Fn(y)48 b Fm(j)p Fn(H)31317 16720 y Fs(N)32213 16521 y Fw(\()p Fn(y)343 b Fw(+)295 b Fn(\025)p Fw(\))p Fm(j)36637 15972 y Fq(2)37162 15046 y Fj(\021)37955 15344 y Fq(1)p Fs(=)p Fq(2)39422 16521 y Fm(k)40187 16185 y Fj(b)40086 16521 y Fw(\010)41025 16720 y Fs(N)41922 16521 y Fm(k)42586 16720 y Fq(2)43112 16521 y Fn(:)3432 20588 y Fw(Since)449 b Fn(H)7922 20787 y Fs(N)9268 20588 y Fw(and)12151 20252 y(~)11814 20588 y Fn(H)12892 20787 y Fs(N)14238 20588 y Fw(deca)-36 b(y)450 b(exp)36 b(onen)-36 b(tially)-108 b(,)455 b(see)450 b(\(6.16\),)455 b(w)-36 b(e)450 b(ha)-36 b(v)g(e)451 b Fm(j)p Fn(D)38815 20787 y Fs(N)39711 20588 y Fw(\()p Fn(x)p Fw(\))p Fm(j)396 b(\024)h Fn(C)95 b(e)45286 20106 y Fi(\000)p Fs(x)3432 22193 y Fw(for)453 b(some)g Fn(C)497 b(>)401 b Fw(0)453 b(indep)36 b(enden)-36 b(t)451 b(of)j Fn(N)139 b Fw(.)635 b(Therefore)454 b(b)-36 b(y)452 b(dominated)h(con)-36 b(v)g(ergence)453 b(w)-36 b(e)3432 23798 y(obtain)5246 27045 y(lim)4787 27872 y Fs(N)94 b Fi(!1)7510 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Fn(a)p Fw(\))28384 32893 y Fi(\000)p Fq(1)29642 33441 y Fn(:)13276 b Fw(\(7.16\))3432 36140 y(Consequen)-36 b(tly)357 b(w)-36 b(e)357 b(need)g(to)g(estimate)g(the) f(in)-36 b(tegral)358 b(close)g(to)f Fn(z)428 b Fw(=)369 b(0)357 b(on)g(\000)40196 36339 y Fq(1)p Fi(\000)p Fs(\032)41935 36140 y Fw(\()p Fn(\016)50 b Fw(\))356 b(only)-108 b(.)3432 37745 y(W)g(e)433 b(use)g(the)g(T)-108 b(a)-36 b(ylor)435 b(expansion,)10526 40963 y Fn(f)11167 41162 y Fs(N)12064 40963 y Fw(\()p Fn(z)59 b Fw(\))1107 b(=)f Fn(f)17605 41162 y Fs(N)18502 40963 y Fw(\(0\))295 b(+)g Fn(f)22549 40414 y Fi(0)22407 41291 y Fs(N)23303 40963 y Fw(\(0\))p Fn(z)355 b Fw(+)27364 40064 y(1)p 27364 40657 V 27364 41874 a(2)28147 40963 y Fn(f)28930 40414 y Fi(00)28788 41291 y Fs(N)29684 40963 y Fw(\(0\))p Fn(z)32009 40414 y Fq(2)32831 40963 y Fw(+)34270 40064 y(1)p 34270 40657 V 34270 41874 a(6)35053 40963 y Fn(f)35836 40414 y Fi(000)35694 41291 y Fs(N)36657 40963 y Fw(\(0\))p Fn(z)38982 40414 y Fq(3)43279 40963 y Fw(\(7.17\))16964 43660 y(+)p Fm(O)19071 42584 y Fj(\000)19679 43660 y Fm(j)p Fn(z)59 b Fm(j)21080 43111 y Fq(4)22308 43660 y Fw(max)21828 44545 y Fq(0)p Fi(\024j)p Fs(z)37 b Fi(j\024)p Fs(\016)25431 43660 y Fm(j)p Fn(f)26583 42981 y Fq(\()p Fs(iv)32 b Fq(\))26441 44020 y Fs(N)28180 43660 y Fw(\()p Fn(z)59 b Fw(\))p Fm(j)30224 42584 y Fj(\001)30833 43660 y Fn(:)12085 b Fw(\(7.18\))3432 47026 y(Some)433 b(computations)g(yield)13939 49725 y Fn(f)14722 49177 y Fi(0)14580 50054 y Fs(N)15477 49725 y Fw(\(0\))1107 b(=)f Fm(\000)p Fn(N)22578 49177 y Fi(\000)p Fq(2)p Fs(=)p Fq(3)24776 49725 y Fw(\()p Fn(y)343 b Fw(+)295 b Fn(s)p Fw(\))p Fn(\024)f Fw(+)h Fm(O)37 b Fw(\()p Fn(N)33817 49177 y Fi(\000)p Fq(1)35073 49725 y Fw(\))p Fn(;)13939 51662 y(f)14722 51114 y Fi(00)14580 51991 y Fs(N)15477 51662 y Fw(\(0\))1107 b(=)f Fn(N)21545 51114 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)23743 51662 y Fw(2)p Fn(w)36 b(\024)26108 51114 y Fq(2)26929 51662 y Fw(+)294 b Fm(O)37 b Fw(\()p Fn(N)31017 51114 y Fi(\000)p Fq(2)p Fs(=)p Fq(3)33215 51662 y Fw(\))p Fn(;)9197 b Fw(\(7.19\))13873 53600 y Fn(f)14656 53051 y Fi(000)14514 53928 y Fs(N)15477 53600 y Fw(\(0\))1107 b(=)f(2)p Fn(\024)21763 53051 y Fq(3)22584 53600 y Fw(+)295 b Fm(O)37 b Fw(\()p Fn(N)26673 53051 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)28870 53600 y Fw(\))p Fn(;)3432 56479 y Fw(and)483 b Fm(j)p Fn(f)7163 55800 y Fq(\()p Fs(iv)32 b Fq(\))7021 56839 y Fs(N)8761 56479 y Fw(\()p Fn(z)59 b Fw(\))p Fm(j)455 b Fw(=)f Fm(O)37 b Fw(\(1\))484 b(for)g Fm(j)p Fn(z)59 b Fm(j)456 b(\024)f Fn(\016)50 b Fw(.)730 b(The)484 b(c)-36 b(hange)484 b(of)h(v)-72 b(ariable)485 b Fn(\034)603 b Fw(=)454 b Fn(N)40415 55997 y Fq(1)p Fs(=)p Fq(3)41881 56479 y Fn(\024z)544 b Fw(leads)3432 58084 y(to)6234 61043 y Fn(N)139 b(f)8056 61242 y Fs(N)8952 61043 y Fw(\()p Fn(z)59 b Fw(\))369 b(=)g Fn(N)139 b(f)14199 61242 y Fs(N)15095 61043 y Fw(\(0\))295 b Fm(\000)g Fn(\034)148 b Fw(\()p Fn(y)343 b Fw(+)295 b Fn(s)p Fw(\))g(+)g Fn(w)36 b(\034)26283 60495 y Fq(2)27103 61043 y Fw(+)28543 60145 y(1)p 28543 60738 V 28543 61955 a(3)29326 61043 y Fn(\034)30039 60495 y Fq(3)30860 61043 y Fw(+)295 b Fm(O)37 b Fw(\()p Fn(\034)76 b(;)221 b(\034)35704 60495 y Fq(4)36230 61043 y Fw(\))p Fn(N)37917 60495 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)40115 61043 y Fn(:)2803 b Fw(\(7.20\))3432 64082 y(Consequen)-36 b(tly)6942 67352 y Fn(e)7545 66804 y Fi(\000)p Fs(N)94 b(f)9570 66953 y Fe(N)10339 66804 y Fq(\(0\))11818 65544 y Fj(Z)12556 68552 y Fq(\000)13144 68675 y Fg(1)p Ff(\000)p Fe(\032)14669 68552 y Fq(\()p Fs(\016)33 b Fq(\))16127 67352 y Fn(e)16730 66804 y Fs(N)94 b(f)18023 66953 y Fe(N)18792 66804 y Fq(\()p Fs(z)37 b Fq(\))20055 67352 y Fw(d)p Fn(z)428 b Fw(=)368 b Fn(\024)23939 66804 y Fi(\000)p Fq(1)25197 67352 y Fn(N)26378 66804 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)28797 65544 y Fj(Z)29535 68552 y Fs(\024N)30918 68290 y Fg(1)p Fe(=)p Fg(3)32210 68552 y Fq(\000)32798 68675 y Fg(1)p Ff(\000)p Fe(\032)34322 68552 y Fq(\()p Fs(\016)33 b Fq(\))35781 67352 y Fw(d)p Fn(\034)148 b(e)37820 66804 y Fi(\000)p Fs(\034)102 b Fq(\()p Fs(y)32 b Fq(+)p Fs(s)p Fq(\)+)p Fs(w)24 b(\034)43419 66491 y Fg(2)43878 66804 y Fq(+)44743 66442 y Fg(1)p 44743 66603 406 40 v 44743 67151 a(3)45282 66804 y Fs(\034)45802 66491 y Fg(3)4823 71055 y Fw(+)1107 b Fn(\024)7691 70506 y Fi(\000)p Fq(1)8948 71055 y Fn(N)10129 70506 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)12549 69246 y Fj(Z)13287 72254 y Fs(\024N)14670 71992 y Fg(1)p Fe(=)p Fg(3)15961 72254 y Fq(\000)16549 72377 y Fg(1)p Ff(\000)p Fe(\032)18074 72254 y Fq(\()p Fs(\016)33 b Fq(\))19532 71055 y Fw(d)p Fn(\034)148 b(e)21571 70506 y Fi(\000)p Fs(\034)102 b Fq(\()p Fs(y)32 b Fq(+)p Fs(s)p Fq(\)+)p Fs(w)24 b(\034)27170 70194 y Fg(2)27630 70506 y Fq(+)28495 70144 y Fg(1)p 28495 70305 V 28495 70853 a(3)29034 70506 y Fs(\034)29554 70194 y Fg(3)30070 69580 y Fj(\020)30863 71055 y Fn(e)31466 70506 y Fi(O)i Fq(\()p Fs(\034)50 b(;\034)33852 70194 y Fg(4)34313 70506 y Fq(\))p Fs(N)35520 70194 y Ff(\000)p Fg(1)p Fe(=)p Fg(3)37802 71055 y Fm(\000)295 b Fw(1)39780 69580 y Fj(\021)40574 71055 y Fn(:)2344 b Fw(\(7.21\))24367 75163 y(32)p eop end end %%Page: 33 33 TeXDict begin HPSdict begin 33 32 bop 3432 7638 a Fw(The)433 b(last)h(term)f(can)h(b)36 b(e)433 b(estimated)h(using)f(that)g Fm(j)p Fn(e)28957 7156 y Fs(x)29837 7638 y Fm(\000)296 b Fw(1)p Fm(j)369 b(\024)g(j)p Fn(x)p Fm(j)p Fn(e)36036 7156 y Fi(j)p Fs(x)p Fi(j)37144 7638 y Fw(,)434 b(i.e.,)g(using)11414 9462 y Fj(\020)12208 10937 y Fn(e)12811 10388 y Fi(O)26 b Fq(\()p Fs(\034)50 b(;\034)15197 10076 y Fg(4)15658 10388 y Fq(\))p Fs(N)16865 10076 y Ff(\000)p Fg(1)p Fe(=)p Fg(3)19146 10937 y Fm(\000)296 b Fw(1)21125 9462 y Fj(\021)22287 10937 y Fw(=)369 b Fn(e)24271 10388 y Fi(O)26 b Fq(\()p Fs(\034)50 b(;\034)26657 10076 y Fg(4)27118 10388 y Fq(\))p Fs(N)28325 10076 y Ff(\000)p Fg(1)p Fe(=)p Fg(3)30311 10937 y Fm(O)37 b Fw(\()p Fn(\034)76 b(;)221 b(\034)33848 10388 y Fq(4)34374 10937 y Fw(\))p Fn(N)36061 10388 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)38259 10937 y Fn(:)4659 b Fw(\(7.22\))3432 14333 y(The)496 b(term)g(in)g(the)g(exp)36 b(onen)-36 b(t)496 b(is)h(of)g(the)f(form)h Fm(\000)p Fn(\034)148 b Fw(\()p Fn(y)386 b Fw(+)338 b Fn(s)p Fw(\))p Fn(\037)33792 14532 y Fq(1)34655 14333 y Fw(+)f Fn(w)36 b(\034)37683 13851 y Fq(2)38209 14333 y Fn(\037)39024 14532 y Fq(2)39888 14333 y Fw(+)41370 13810 y Fq(1)p 41370 14027 471 54 v 41370 14791 a(3)41973 14333 y Fn(\034)42686 13851 y Fq(3)43212 14333 y Fn(\037)44027 14532 y Fq(3)45049 14333 y Fw(for)3432 15938 y(some)408 b Fn(\037)7480 16137 y Fq(1)8005 15938 y Fn(;)221 b(\037)9402 16137 y Fq(2)9928 15938 y Fn(;)g(\037)11325 16137 y Fq(3)11852 15938 y Fw(.)570 b(By)408 b(taking)g Fn(\016)458 b Fw(small)408 b(enough)f Fn(\037)28462 16137 y Fq(1)28988 15938 y Fn(;)221 b(\037)30385 16137 y Fq(2)30911 15938 y Fn(;)g(\037)32308 16137 y Fq(3)33242 15938 y Fw(can)408 b(b)36 b(e)407 b(made)g(as)h(close)h(to)3432 17543 y(1)383 b(as)h(desired.)561 b(Th)-36 b(us)383 b(the)g(second)g(in)-36 b(tegral)384 b(con)-36 b(v)g(erges)384 b(and)f(the)g(error)h(term)f(in)g(\(7.21\)) 3432 19148 y(is)433 b(of)i(order)e Fm(O)37 b Fw(\()p Fn(N)12399 18666 y Fi(\000)p Fq(2)p Fs(=)p Fq(3)14596 19148 y Fw(\).)5383 20753 y(Finally)434 b(w)-36 b(e)434 b(estimate)g(the)f(leading)h(term)13273 24259 y Fn(\024)14022 23710 y Fi(\000)p Fq(1)15279 24259 y Fn(N)16460 23710 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)18879 22451 y Fj(Z)19617 25458 y Fs(\024N)21000 25197 y Fg(1)p Fe(=)p Fg(3)22292 25458 y Fq(\000)22880 25581 y Fg(1)p Ff(\000)p Fe(\032)24405 25458 y Fq(\()p Fs(\016)33 b Fq(\))25863 24259 y Fw(d)p Fn(\034)148 b(e)27902 23710 y Fi(\000)p Fs(\034)102 b Fq(\()p Fs(y)32 b Fq(+)p Fs(s)p Fq(\)+)p Fs(w)24 b(\034)33501 23398 y Fg(2)33961 23710 y Fq(+)34826 23348 y Fg(1)p 34826 23509 406 40 v 34826 24058 a(3)35364 23710 y Fs(\034)35884 23398 y Fg(3)36401 24259 y Fn(:)6517 b Fw(\(7.23\))3432 28295 y(Deforming)297 b(the)e(in)-36 b(tegration)296 b(path)g(from)g(\000\()p Fn(\016)50 b Fw(\))p Fn(N)27710 27813 y Fq(1)p Fs(=)p Fq(3)29175 28295 y Fn(\024)296 b Fw(to)g(\000\()p Fn(\016)50 b Fw(\))p Fn(N)35304 27813 y Fq(1)p Fs(=)p Fq(3)36769 28295 y Fn(\024)14 b Fm(\000)g Fn(w)36 b Fw(,)324 b(one)296 b(obtains)11354 31875 y Fn(e)11957 31326 y Fq(\()p Fs(y)32 b Fq(+)p Fs(s)p Fq(\))p Fs(w)15107 31875 y Fn(e)15843 30964 y Fg(2)p 15843 31125 V 15843 31673 a(3)16382 31326 y Fs(w)17081 31014 y Fg(3)17597 31875 y Fn(\024)18346 31326 y Fi(\000)p Fq(1)19604 31875 y Fn(N)20785 31326 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)23204 30067 y Fj(Z)23942 33074 y Fq(\000\()p Fs(\016)h Fq(\))p Fs(N)26552 32812 y Fg(1)p Fe(=)p Fg(3)27843 33074 y Fs(\024)28662 31875 y Fw(d)p Fn(\034)148 b(e)30701 31326 y Fi(\000)p Fs(\034)102 b Fq(\()p Fs(y)32 b Fq(+)p Fs(s)p Fq(+)p Fs(w)35414 31014 y Fg(2)35874 31326 y Fq(\)+)37105 30964 y Fg(1)p 37105 31125 V 37105 31673 a(3)37644 31326 y Fs(\034)38164 31014 y Fg(3)43279 31875 y Fw(\(7.24\))3432 35880 y(up)448 b(to)h(an)h Fm(O)37 b Fw(\()p Fn(e)10958 35398 y Fi(\000)p Fs(\026N)13151 35880 y Fw(\))449 b(error.)625 b(By)450 b(extending)f(the)g(in)-36 b(tegral)450 b(to)f Fn(e)35085 35398 y Fi(\006)p Fs(i\031)32 b(=)p Fq(3)37705 35880 y Fm(1)450 b Fw(one)f(pic)-36 b(ks)450 b(up)3432 37486 y(an)433 b(error)g(at)h(most)g Fm(O)37 b Fw(\()p Fn(e)15398 37003 y Fi(\000)p Fs(\026N)17591 37486 y Fw(\),)434 b(again.)579 b(But)16519 39257 y Fj(Z)17257 42264 y Fq(\000\()p Fi(1)p Fq(\))19795 41065 y Fw(d)p Fn(z)59 b(e)21917 40154 y Fg(1)p 21916 40315 V 21916 40864 a(3)22455 40516 y Fs(z)22930 40204 y Fg(3)23391 40516 y Fi(\000)p Fs(xz)25552 41065 y Fw(=)369 b Fm(\000)p Fw(2)p Fn(\031)48 b(i)221 b Fw(Ai\()p Fn(x)p Fw(\))p Fn(;)9764 b Fw(\(7.25\))3432 45071 y(where)339 b(\000\()p Fm(1)p Fw(\))g(is)g(the)g(path)f(joining)j (0)e(with)g Fn(e)25552 44589 y Fi(\006)p Fs(i\031)32 b(=)p Fq(3)28173 45071 y Fm(1)339 b Fw(b)-36 b(y)340 b(straigh)-36 b(t)339 b(lines)g(orien)-36 b(ted)339 b(with)3432 46676 y(imaginary)g(part)g(decreasing.)546 b(Note)339 b(that)f(in)h(\(D.2\))g(the)f(orien)-36 b(tation)339 b(is)g(the)f(opp)36 b(osite.)3432 48281 y(The)402 b(error)g(term)g Fm(O)37 b Fw(\()p Fn(e)14502 47799 y Fi(\000)p Fs(\026N)16695 48281 y Fw(\))402 b(can)h(b)36 b(e)402 b(b)36 b(ounded)400 b(b)-36 b(y)403 b Fm(O)37 b Fw(\()p Fn(N)31533 47799 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)33730 48281 y Fw(\).)567 b(Th)-36 b(us)402 b(putting)f(all)j(the)3432 49886 y(terms)433 b(together)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(pro)-36 b(v)g(ed)433 b(that)g(for)h(an)-36 b(y)434 b Fo(\014xe)-66 b(d)570 b Fn(y)3705 52819 y(I)4279 53026 y Fs(N)94 b Fq(+)p Fs(\013=)p Fq(2)p Fs(;\013=)p Fq(2)9258 52819 y Fw(\()p Fn(u)296 b Fw(+)f Fn(y)48 b(\024N)14719 52271 y Fq(1)p Fs(=)p Fq(3)16184 52819 y Fw(\))p Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))p Fn(\024N)21300 52271 y Fq(1)p Fs(=)p Fq(3)23873 52819 y Fw(=)1106 b Fm(\000)p Fw(2)p Fn(\031)48 b(i)221 b Fw(Ai)q(\()p Fn(w)31934 52271 y Fq(2)32754 52819 y Fw(+)295 b Fn(s)g Fw(+)g Fn(y)48 b Fw(\))p Fn(e)38067 52271 y Fs(w)24 b Fq(\()p Fs(w)39831 51958 y Fg(2)40291 52271 y Fq(+)p 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y(W)-108 b(e)318 b(ha)-36 b(v)g(e)320 b(sho)-36 b(wn)318 b(that)h(for)g(all)h Fn(\032)369 b Fm(2)f Fw(\(0)p Fn(;)221 b Fw(1\),)26515 18376 y Fq(d)p Fs(R)5 b(e)p Fq(\()p Fs(f)29006 18525 y Fe(N)29776 18376 y Fq(\))p 26515 18703 3627 54 v 27897 19467 a(d)p Fs(t)30643 19009 y Fn(<)369 b Fw(0)319 b(for)h Fn(t)368 b(>)h Fw(0,)38571 18376 y Fq(d)p Fs(R)5 b(e)p Fq(\()p Fs(f)41062 18525 y Fe(N)41832 18376 y Fq(\))p 38571 18703 V 39953 19467 a(d)p Fs(t)42700 19009 y Fn(>)368 b Fw(0)319 b(for)3432 20614 y Fn(t)368 b(<)h Fw(0)345 b(for)f Fn(L)369 b Fm(\035)h Fw(1,)362 b Fn(")369 b Fm(\034)g Fw(1,)363 b(and)344 b Fn(N)483 b Fw(large)345 b(enough.)548 b(Th)-36 b(us)344 b Fn(\015)32809 20813 y Fq(1)33679 20614 y Fw(is)g(a)h(steep)f(descen) -36 b(t)343 b(path.)3432 22219 y(Therefore,)434 b(if)g(w)-36 b(e)434 b(denote)f(b)-36 b(y)434 b(\000)19600 22418 y Fq(1)p Fi(\000)p Fs(\032)21339 22219 y Fw(\()p Fn(\016)50 b Fw(\))22977 21737 y Fs(c)23873 22219 y Fw(the)433 b(p)36 b(ortion)433 b(of)i(\000)32996 22418 y Fq(1)p Fi(\000)p Fs(\032)35168 22219 y Fw(with)f Fm(j)p Fn(z)354 b Fm(\000)296 b Fn(z)41391 22418 y Fq(0)41917 22219 y Fm(j)369 b Fn(>)f(\016)50 b Fw(,)13926 24029 y Fj(\014)13926 24826 y(\014)13926 25623 y(\014)13926 26420 y(\014)14591 24147 y(Z)15328 27155 y Fq(\000)15916 27278 y Fg(1)p Ff(\000)p Fe(\032)17441 27155 y Fq(\()p Fs(\016)33 b Fq(\))18622 26903 y Fe(c)19326 25956 y Fw(d)p Fn(z)59 b(e)21315 25407 y Fs(N)94 b(f)22608 25556 y Fe(N)23377 25407 y Fq(\()p Fs(z)37 b Fq(\))24640 24029 y Fj(\014)24640 24826 y(\014)24640 25623 y(\014)24640 26420 y(\014)25451 25956 y Fm(\024)370 b Fn(e)27457 25407 y Fs(N)94 b(f)28750 25556 y Fe(N)29519 25407 y Fq(\()p Fs(z)30323 25530 y Fg(0)30784 25407 y Fq(\))31205 25956 y Fm(O)37 b Fw(\()p Fn(e)33409 25407 y Fi(\000)p Fs(\026N)35602 25956 y Fw(\))7171 b(\(7.33\))3432 29768 y(for)434 b(some)g Fn(\026)368 b(>)h Fw(0.)5383 31373 y(Finally)552 b(w)-36 b(e)552 b(ha)-36 b(v)g(e)552 b(to)f(ev)-72 b(aluate)552 b(the)f(con)-36 b(tribution)551 b(coming)h(from)g(\000)40478 31572 y Fq(1)p Fi(\000)p Fs(\032)42217 31373 y Fw(\()p Fn(\016)50 b Fw(\),)581 b(the)3432 32979 y(p)36 b(ortion)594 b(of)h(\000)10636 33178 y Fq(1)p Fi(\000)p Fs(\032)12969 32979 y Fw(with)g Fm(j)p Fn(z)464 b Fm(\000)405 b Fn(z)19572 33178 y Fq(0)20098 32979 y Fm(j)642 b(\024)h Fn(\016)50 b Fw(.)1060 b(The)595 b(con)-36 b(tribution)593 b(of)i(the)f(part)g(in) g Fn(\015)44609 33178 y Fq(1)45729 32979 y Fw(is)3432 34584 y(estimated)433 b(as)h(follo)-36 b(ws.)580 b(On)433 b Fn(\015)18580 34783 y Fq(1)19106 34584 y Fw(,)13574 38232 y Fn(R)11 b(e)p Fw(\()p Fn(f)16325 38431 y Fs(N)17222 38232 y Fw(\()p Fn(z)59 b Fw(\)\))368 b(=)h Fn(f)21793 38431 y Fs(N)22690 38232 y Fw(\()p Fn(z)23800 38431 y Fq(0)24326 38232 y Fw(\))295 b Fm(\000)26588 37334 y Fn(t)27058 36852 y Fq(2)p 26588 37927 996 54 v 26760 39144 a Fw(2)27716 38232 y Fn(f)28499 37684 y Fi(00)28357 38561 y Fs(N)29254 38232 y Fw(\()p Fn(z)30364 38431 y Fq(0)30889 38232 y Fw(\))g(+)g Fm(O)37 b Fw(\()p Fn(t)35068 37684 y Fq(4)35593 38232 y Fw(\))p Fn(:)6819 b Fw(\(7.34\))3432 41729 y(Some)1343 b(computations)g(leads)g(to)h Fn(f)24271 41247 y Fi(00)24129 42064 y Fs(N)25026 41729 y Fw(\()p Fn(z)26136 41928 y Fq(0)26661 41729 y Fw(\))1918 b(=)f Fm(O)37 b Fw(\()p Fn(w)f(N)35762 41247 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)37959 41729 y Fw(\))914 b(+)g(\(2)p Fn(\024)43210 41247 y Fq(3)p Fs(=)p Fq(2)45591 41729 y Fw(+)3432 43334 y Fm(O)37 b Fw(\()p Fn(w)f(N)7180 42852 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)9376 43334 y Fw(\)\))88 b(~)-738 b Fn(y)11070 42852 y Fq(1)p Fs(=)p Fq(2)12935 43334 y Fw(+)399 b Fm(O)37 b Fw(\()88 b(~)-738 b Fn(y)47 b Fw(\))p Fn(:)586 b Fw(F)-108 b(or)585 b Fn(L)628 b Fm(\035)h Fw(1,)624 b Fn(")k Fm(\034)g Fw(1,)d(and)585 b Fn(N)724 b Fw(large)587 b(enough,)624 b(it)585 b(fol-)3432 44939 y(lo)-36 b(ws)584 b(that)e Fn(f)10193 44457 y Fi(00)10051 45275 y Fs(N)10948 44939 y Fw(\()p Fn(z)12058 45138 y Fq(0)12584 44939 y Fw(\))623 b Fm(\025)15503 44416 y Fq(3)p 15503 44634 471 54 v 15503 45397 a(2)16106 44939 y Fn(\024)16855 44457 y Fq(3)p Fs(=)p Fq(2)18410 44939 y Fw(~)-738 b Fn(y)19004 44457 y Fq(1)p Fs(=)p Fq(2)20470 44939 y Fw(.)1028 b(On)582 b(the)h(other)f(hand,)620 b(the)583 b(term)g(prop)36 b(ortional)3432 46713 y(to)523 b Fn(t)5581 46231 y Fq(4)6630 46713 y Fw(is)h(m)-36 b(uc)g(h)523 b(smaller)h(than)f(the)g Fn(t)22089 46231 y Fq(2)23138 46713 y Fw(term.)848 b(In)524 b(fact,)547 b(for)524 b(0)e Fm(\024)h Fn(t)f Fm(\024)39128 45614 y(p)p 40235 45614 651 54 v 1099 x Fw(3)p Fn(z)41489 46912 y Fq(0)42538 46713 y Fm(\030)610 b Fw(~)-738 b Fn(y)44775 46231 y Fq(1)p Fs(=)p Fq(2)46242 46713 y Fw(,)3432 48318 y Fn(t)3902 47836 y Fq(2)4796 48318 y Fm(\024)457 b Fw(~)-738 b Fn(y)48 b Fm(O)37 b Fw(\(1\))368 b Fm(\024)458 b Fw(~)-738 b Fn(y)12090 47836 y Fq(1)p Fs(=)p Fq(2)13556 48318 y Fm(O)37 b Fw(\()p Fn(")15766 47836 y Fq(1)p Fs(=)p Fq(2)17232 48318 y Fw(\))369 b Fm(\034)g Fn(f)20587 47836 y Fi(00)20445 48654 y Fs(N)21342 48318 y Fw(\()p Fn(z)22452 48517 y Fq(0)22977 48318 y Fw(\).)579 b(Therefore)3585 50093 y Fj(\014)3585 50890 y(\014)3585 51687 y(\014)3585 52484 y(\014)4249 50211 y(Z)4987 53218 y Fs(\015)5472 53341 y Fg(1)6210 52019 y Fw(d)p Fn(z)59 b(e)8199 51470 y Fs(N)94 b(f)9492 51619 y Fe(N)10261 51470 y Fq(\()p Fs(z)37 b Fq(\))11524 50093 y Fj(\014)11524 50890 y(\014)11524 51687 y(\014)11524 52484 y(\014)13073 52019 y Fm(\024)1107 b Fn(e)15816 51470 y Fs(N)94 b(f)17109 51619 y Fe(N)17879 51470 y Fq(\()p Fs(z)18683 51593 y Fg(0)19144 51470 y Fq(\))19786 50211 y Fj(Z)20524 53218 y Fk(R)21582 52019 y Fw(d)p Fn(te)23378 51470 y Fi(\000)24243 51108 y Fg(1)p 24242 51269 406 40 v 24242 51818 a(3)24781 51470 y Fs(t)25121 51158 y Fg(2)25582 51470 y Fs(f)26132 51158 y Ff(00)26034 51771 y Fe(N)26803 51470 y Fq(\()p Fs(z)27607 51593 y Fg(0)28068 51470 y Fq(\))p Fs(N)29699 52019 y Fm(\024)369 b Fn(e)31704 51470 y Fs(N)94 b(f)32997 51619 y Fe(N)33767 51470 y Fq(\()p Fs(z)34571 51593 y Fg(0)35032 51470 y Fq(\))35674 50211 y Fj(Z)36412 53218 y Fk(R)37470 52019 y Fw(d)p Fn(te)39266 51470 y Fi(\000)40131 51108 y Fg(1)p 40130 51269 V 40130 51818 a(2)40669 51470 y Fs(t)41009 51158 y Fg(2)41470 51470 y Fs(\024)42012 51158 y Fg(2)42473 51470 y Fs(y)42970 51158 y Fg(1)p Fe(=)p Fg(2)44261 51470 y Fs(N)45102 51158 y Fg(2)p Fe(=)p Fg(3)13084 55046 y Fw(=)1117 b Fn(e)15816 54497 y Fs(N)94 b(f)17109 54646 y Fe(N)17879 54497 y Fq(\()p Fs(z)18683 54620 y Fg(0)19144 54497 y Fq(\))19565 55046 y Fn(y)20247 54497 y Fi(\000)p Fq(1)p Fs(=)p Fq(4)22445 55046 y Fn(N)23626 54497 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)25824 55046 y Fm(O)37 b Fw(\(1\))p Fn(:)14337 b Fw(\(7.35\))5383 57979 y(In)459 b(\(7.33\))i(and)e (\(7.35\))i(w)-36 b(e)460 b(still)g(ha)-36 b(v)g(e)460 b(to)g(ev)-72 b(aluate)460 b Fn(f)32062 58178 y Fs(N)32959 57979 y Fw(\()p Fn(z)34069 58178 y Fq(0)34595 57979 y Fw(\))313 b Fm(\000)g Fn(f)37401 58178 y Fs(N)38297 57979 y Fw(\(0\).)657 b(A)460 b(compu-)3432 59584 y(tation)433 b(leads)h(to)10834 62778 y Fn(N)139 b(f)12656 62977 y Fs(N)13552 62778 y Fw(\()p Fn(z)14662 62977 y Fq(0)15188 62778 y Fw(\))295 b Fm(\000)g Fn(N)139 b(f)19139 62977 y Fs(N)20035 62778 y Fw(\(0\))369 b(=)g Fm(\000)p Fn(sy)25775 62229 y Fq(1)p Fs(=)p Fq(2)27241 62778 y Fn(\037)28056 62977 y Fq(1)28877 62778 y Fw(+)295 b Fn(w)36 b(y)48 b(\037)32647 62977 y Fq(2)33467 62778 y Fm(\000)34928 61879 y Fw(2)p 34928 62472 651 54 v 34928 63689 a(3)35711 62778 y Fn(y)36393 62229 y Fq(3)p Fs(=)p Fq(2)37860 62778 y Fn(\037)38675 62977 y Fq(3)43279 62778 y Fw(\(7.36\))3432 66051 y(for)420 b(some)h Fn(\037)9466 66250 y Fq(1)9992 66051 y Fn(;)221 b(\037)11389 66250 y Fq(2)11915 66051 y Fn(;)g(\037)13312 66250 y Fq(3)14259 66051 y Fw(whic)-36 b(h)420 b(can)g(b)36 b(e)420 b(made)g(as)h(close)g(to)f(1)h(as)g(w)-36 b(an)g(ted)419 b(b)-36 b(y)420 b(c)-36 b(ho)36 b(osing)421 b Fn(")3432 67656 y Fw(small)434 b(enough.)578 b(On)433 b(the)g(other)g(hand,)g(since)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b Fn(y)417 b Fm(\025)369 b Fn(L)p Fw(,)434 b(for)g Fn(L)f Fw(large)i(enough,)16471 71108 y Fn(N)139 b(f)18293 71307 y Fs(N)19189 71108 y Fw(\()p Fn(z)20299 71307 y Fq(0)20824 71108 y Fw(\))295 b Fm(\000)h Fn(N)139 b(f)24776 71307 y Fs(N)25672 71108 y Fw(\(0\))369 b Fm(\024)g(\000) 30271 70209 y Fw(1)p 30271 70802 V 30271 72019 a(3)31054 71108 y Fn(y)31736 70559 y Fq(3)p Fs(=)p Fq(2)33203 71108 y Fn(:)9715 b Fw(\(7.37\))24367 75163 y(35)p eop end end %%Page: 36 36 TeXDict begin HPSdict begin 36 35 bop 3432 7638 a Fw(Th)-36 b(us)433 b(the)g(con)-36 b(tributions)432 b(\(7.33\))j(and)e(\(7.35\))h (can)g(b)36 b(e)433 b(b)36 b(ounded)432 b(b)-36 b(y)13048 10768 y(\(7)p Fn(:)p Fw(33\))1108 b Fm(\024)f Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))22299 10220 y Fi(\000)p Fq(1)23557 10768 y Fn(e)24160 10220 y Fi(\000)p Fs(\026N)26354 10768 y Fn(e)26957 10220 y Fi(\000)27822 9858 y Fg(1)p 27821 10019 406 40 v 27821 10567 a(3)28360 10220 y Fs(y)28857 9907 y Fg(3)p Fe(=)p Fg(2)30204 10768 y Fm(O)37 b Fw(\(1\))p Fn(;)13048 12902 y Fw(\(7)p Fn(:)p Fw(35\))1108 b Fm(\024)f Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))22299 12354 y Fi(\000)p Fq(1)23557 12902 y Fn(N)24738 12354 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)26936 12902 y Fn(e)27539 12354 y Fi(\000)28404 11992 y Fg(1)p 28403 12153 V 28403 12701 a(3)28942 12354 y Fs(y)29439 12041 y Fg(3)p Fe(=)p Fg(2)30785 12902 y Fn(L)31670 12354 y Fi(\000)p Fq(1)p Fs(=)p Fq(4)33869 12902 y Fm(O)37 b Fw(\(1\))p Fn(:)6292 b Fw(\(7.38\))5383 15836 y(The)571 b(\014nal)h(step)f(is)h(to)g(b)36 b(ound)571 b(the)g(con)-36 b(tribution)571 b(coming)h(form)g Fn(\015)39862 16035 y Fq(2)40777 15836 y Fm([)389 b Fw(\000)42865 16035 y Fq(1)p Fi(\000)p Fs(\032)44604 15836 y Fw(\()p Fn(\016)50 b Fw(\).)3432 17441 y(F)-108 b(rom)433 b(the)g(asymptotics)h(of)g (\014xed)f Fn(y)482 b Fw(one)433 b(has,)h(using)g Fn(z)428 b Fw(=)369 b Fn(te)33985 16959 y Fs(i\031)32 b(=)p Fq(3)35873 17441 y Fw(,)5676 20854 y Fn(R)11 b(e)p Fw(\()p Fn(f)8427 21053 y Fs(N)9323 20854 y Fw(\()p Fn(z)59 b Fw(\)\))1107 b(=)f Fn(f)15370 21053 y Fs(N)16267 20854 y Fw(\(0\))295 b Fm(\000)g Fw(\()88 b(~)-738 b Fn(y)343 b Fw(+)295 b Fn(s\024N)24885 20305 y Fi(\000)p Fq(2)p Fs(=)p Fq(3)27082 20854 y Fw(\))27811 19955 y Fn(t)p 27721 20548 651 54 v 27721 21765 a Fw(2)28799 20854 y(+)g Fm(O)37 b Fw(\()p Fn(tN)33358 20305 y Fi(\000)p Fq(1)34614 20854 y Fw(\))8159 b(\(7.39\))14729 24159 y(+)p Fn(w)36 b(\024)17456 23611 y Fq(2)17981 24159 y Fn(N)19162 23611 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)21493 23261 y Fn(t)21963 22779 y Fq(2)p 21493 23854 996 54 v 21665 25070 a Fw(2)22916 24159 y(+)295 b Fm(O)37 b Fw(\()p Fn(t)26294 23611 y Fq(2)26819 24159 y Fn(N)28000 23611 y Fi(\000)p Fq(2)p Fs(=)p Fq(3)30198 24159 y Fw(\))295 b Fm(\000)32460 23261 y Fn(\024)p 32460 23854 749 54 v 32509 25070 a Fw(3)33341 24159 y Fn(t)33811 23611 y Fq(3)34632 24159 y Fw(+)g Fm(O)37 b Fw(\()p Fn(t)38010 23611 y Fq(3)38535 24159 y Fn(N)39716 23611 y Fi(\000)p Fq(1)p Fs(=)p Fq(3)41914 24159 y Fn(;)221 b(t)42966 23611 y Fq(4)43492 24159 y Fw(\))p Fn(:)3432 27715 y Fw(In)587 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Fq(3)p Fs(=)p Fq(2)27320 32530 y Fn(N)28501 32048 y Fi(\000)p Fq(1)29757 32530 y Fw(.)783 b(Th)-36 b(us)501 b(for)i(large)f Fn(L)p Fw(,)519 b(the)501 b(cubic)3432 34135 y(term)456 b(dominates)h(the)g(quadratic)g (one.)649 b(But)456 b(since)i Fn(t)408 b Fm(\024)h Fn(\016)50 b(=)33285 33036 y Fm(p)p 34393 33036 V 34393 34135 a Fw(3,)463 b(the)457 b(quartic)g(term)g(is)3432 35740 y(also)434 b(dominated)f(b)-36 b(y)434 b(the)f(cubic)g(one.)578 b(Therefore,)7759 39482 y Fn(R)11 b(e)p Fw(\()p Fn(f)10510 39681 y Fs(N)11406 39482 y Fw(\()p Fn(z)59 b Fw(\)\))369 b Fm(\024)g Fn(f)15999 39681 y Fs(N)16895 39482 y Fw(\(0\))296 b Fm(\000)20402 38584 y Fw(~)-738 b Fn(y)48 b(t)p 20314 39177 1152 54 v 20564 40393 a Fw(4)21893 39482 y Fm(\000)23355 38584 y Fn(\024t)24574 38102 y Fq(3)p 23355 39177 1745 54 v 23901 40393 a Fw(6)25601 39482 y Fm(\024)369 b Fn(f)27644 39681 y Fs(N)28540 39482 y Fw(\(0\))295 b Fm(\000)32405 38584 y Fw(~)-738 b Fn(y)32999 38102 y Fq(3)p Fs(=)p Fq(2)p 31959 39177 2866 54 v 31959 40393 a Fw(2)p Fn(\024)33358 40010 y Fq(3)p Fs(=)p Fq(2)35252 39482 y Fm(\000)36714 38584 y Fn(\024t)37933 38102 y Fq(3)p 36714 39177 1745 54 v 37261 40393 a Fw(6)38591 39482 y Fn(:)4327 b Fw(\(7.40\))3432 42755 y(Th)-36 b(us)3982 44467 y Fj(\014)3982 45264 y(\014)3982 46061 y(\014)3982 46858 y(\014)4647 44585 y(Z)5385 47593 y Fs(\015)5870 47716 y Fg(2)6330 47593 y Fi([)p Fq(\000)7545 47716 y Fg(1)p Ff(\000)p Fe(\032)9071 47593 y Fq(\()p Fs(\016)33 b Fq(\))8954 46394 y Fw(d)p Fn(z)59 b(e)10943 45845 y Fs(N)94 b(f)12236 45994 y Fe(N)13005 45845 y Fq(\()p Fs(z)37 b Fq(\))14267 44467 y Fj(\014)14267 45264 y(\014)14267 46061 y(\014)14267 46858 y(\014)15079 46394 y Fm(\024)369 b Fw(2)p Fn(e)17734 45845 y Fs(N)94 b(f)19027 45994 y Fe(N)19797 45845 y Fq(\(0\))21055 46394 y Fn(e)21658 45845 y Fi(\000)22523 45483 y Fg(1)p 22522 45644 406 40 v 22522 46192 a(2)23061 45845 y Fs(y)23558 45533 y Fg(3)p Fe(=)p Fg(2)25126 44585 y Fj(Z)26454 44936 y Fi(1)25864 47593 y Fq(0)27672 46394 y Fw(d)p 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b(with)h(the)f(result)4096 59634 y Fn(f)4737 59833 y Fs(N)5633 59634 y Fw(\()p Fn(z)59 b Fw(\))369 b(=)g Fn(f)9699 59833 y Fs(N)10596 59634 y Fw(\()p Fn(z)59 b Fw(\))12271 58505 y Fj(\014)12271 59302 y(\014)12779 60154 y Fq(~)-536 b Fs(y)32 b Fq(=)p Fs("\024=)p Fq(2)16077 59634 y Fm(\000)162 b Fn(z)59 b Fw(\()88 b(~)-738 b Fn(y)210 b Fm(\000)162 b Fn("\024=)p Fw(2\))368 b Fm(\024)i Fn(f)26056 59833 y Fs(N)26952 59634 y Fw(\()p Fn(z)59 b Fw(\))28627 58505 y Fj(\014)28627 59302 y(\014)29135 60154 y Fq(~)-535 b Fs(y)32 b Fq(=)p Fs("\024=)p Fq(2)32433 59634 y Fm(\000)162 b Fw(2)34278 59086 y Fi(\000)p Fq(3)p Fs(=)p Fq(2)36477 58612 y Fm(p)p 37584 58612 609 54 v 1022 x Fn("y)48 b(N)40056 59086 y Fi(\000)p Fq(2)p Fs(=)p Fq(3)42254 59634 y Fn(;)664 b Fw(\(7.42\))3432 63057 y(b)36 b(ecause)433 b Fn(z)428 b Fm(\025)10679 61921 y Fj(p)p 12007 61921 1910 54 v 1136 x Fn("=)p Fw(2)p Fn(\024)14665 62575 y Fi(\000)p Fq(1)16356 63057 y Fw(and)521 b(~)-738 b Fn(y)343 b Fm(\000)296 b Fn("\024=)p 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Fw(\(7.43\))p 45663 70668 45 878 v 45708 69835 781 45 v 45708 70668 V 46487 70668 45 878 v 24367 75163 a(36)p eop end end %%Page: 37 37 TeXDict begin HPSdict begin 37 36 bop 3432 7638 a Fo(Pr)-66 b(o)g(of)601 b(of)i(\(7.4\),)637 b(\(7.5\),)g(and)602 b(\(7.6\).)1028 b Fw(The)583 b(asymptotics)h(of)35216 7302 y(~)35058 7638 y Fn(I)687 b Fw(are)584 b(similar)g(to)g(the)3432 9243 y(one)619 b(of)i Fn(I)104 b Fw(.)1137 b(Instead)620 b(of)g(computing)g(ev)-36 b(erything)620 b(again,)667 b(w)-36 b(e)620 b(sho)-36 b(w)620 b(that,)667 b(via)621 b(the)3432 10848 y(transformation)531 b Fn(\032)k Fm(7!)h Fw(1)362 b Fm(\000)g Fn(\032)531 b Fw(and)g Fn(w)571 b Fm(7!)535 b(\000)p Fn(w)36 b Fw(,)556 b(one)531 b(obtains)g(essen)-36 b(tially)533 b(the)e(same)3432 12453 y(in)-36 b(tegrals)513 b(as)f(already)i(studied)d(for)j Fn(I)104 b Fw(.)815 b(More)513 b(precisely)-108 b(,)532 b(w)-36 b(e)513 b(ha)-36 b(v)g(e)513 b(to)g(estimate)g(the)3432 14058 y(asymptotics)434 b(of)3936 16992 y Fn(W)181 b Fw(\()p Fn(N)66 b(;)221 b(\032;)g(w)36 b Fw(\))1107 b(=)f Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))p Fn(\024N)18101 16443 y Fq(1)p Fs(=)p Fq(3)19567 16992 y Fn(I)20141 17198 y Fs(N)f Fq(+)p Fs(\013=)p Fq(2)p Fs(;\013=)p Fq(2)25121 16992 y Fw(\()p Fn(u)295 b Fw(+)g Fn(y)48 b(\024N)30581 16443 y Fq(1)p Fs(=)p Fq(3)32046 16992 y Fw(\))10727 b(\(7.44\))11373 21026 y(=)13491 19218 y Fj(Z)14229 22226 y Fq(\000)14817 22349 y Fg(1)p Ff(\000)p Fe(\032)16619 21026 y Fw(d)p Fn(z)59 b(e)18608 20478 y Fi(\000)p Fs(z)37 b Fq(\()p Fs(u)p Fq(\()p Fs(N)42 b(;\032;w)24 b Fq(\)+)p Fs(y)32 b(\024)p Fq(\()p Fs(\032)p Fq(\))p Fs(N)27776 20165 y Fg(1)p Fe(=)p Fg(3)29067 20478 y Fq(\))29488 21026 y Fn(\024)p Fw(\()p Fn(\032)p Fw(\))p Fn(N)33101 20478 y Fq(1)p Fs(=)p Fq(3)37591 18170 y Fj(\020)38384 19645 y Fw(1)296 b(+)40773 19122 y Fs(z)p 40770 19339 482 54 v 40770 20103 a(\032)41384 18170 y Fj(\021)42178 18468 y Fs(N)p 34699 20721 11268 54 v 34699 21247 a Fj(\020)35492 22721 y Fw(1)g Fm(\000)38503 22198 y Fs(z)p 37899 22416 1685 54 v 37899 23180 a Fq(1)p Fi(\000)p Fs(\032)39716 21247 y Fj(\021)40509 21544 y Fs(N)94 b Fq(+)p Fs(\013)p Fq(\()p Fs(N)42 b(;\032;w)24 b Fq(\))3432 25946 y Fw(and)4364 28544 y Fj(f)4324 28880 y Fn(W)180 b Fw(\()p Fn(N)66 b(;)221 b(\032;)g(w)36 b Fw(\))1108 b(=)e Fn(Z)95 b Fw(\()p Fn(a)p Fw(\))16559 28331 y Fi(\000)p Fq(1)17817 28880 y Fn(\024N)19747 28331 y Fq(1)p Fs(=)p Fq(3)21371 28544 y Fw(~)21213 28880 y Fn(I)21787 29087 y Fs(N)f Fq(+)p Fs(\013=)p Fq(2)p Fs(;\013=)p Fq(2)26767 28880 y Fw(\()p Fn(u)295 b Fw(+)g Fn(y)48 b(\024N)32227 28331 y Fq(1)p Fs(=)p Fq(3)33692 28880 y Fw(\))9081 b(\(7.45\))11761 32973 y(=)13879 31165 y Fj(Z)14617 34173 y Fq(\000)15205 34292 y Ff(\000)p Fe(\032)16601 32973 y Fw(d)p Fn(z)59 b(e)18590 32425 y Fs(z)37 b Fq(\()p Fs(u)p Fq(\()p Fs(N)42 b(;\032;w)24 b Fq(\)+)p Fs(y)32 b(\024)p Fq(\()p Fs(\032)p Fq(\))p Fs(N)27026 32112 y Fg(1)p Fe(=)p Fg(3)28317 32425 y Fq(\))28738 32973 y Fn(\024)p Fw(\()p Fn(\032)p Fw(\))p Fn(N)32351 32425 y Fq(1)p Fs(=)p 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b(parts)g(to)g(eliminate)h(the)e(terms)3829 41795 y Fs(@)p 3564 42012 1078 54 v 3564 42776 a(@)52 b(x)4775 42318 y Fn(\032)5446 42517 y Fs(s)5936 42318 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))434 b(and)12295 41795 y Fs(@)p 12047 42012 1045 54 v 12047 42776 a(@)52 b(y)13225 42318 y Fn(\032)13896 42517 y Fs(s)14386 42318 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\),)434 b(whic)-36 b(h)433 b(requires)h(the)f(deriv)-72 b(ativ)-36 b(es)8728 45309 y Fn(@)p 8358 45903 1498 54 v 8358 47119 a(@)72 b(x)10126 45872 y Fj(b)9989 46208 y Fw(\011)11001 46407 y Fs(w)11755 46208 y Fw(\()p Fn(x)p Fw(\))1107 b(=)f(Ai)q(\()p Fn(x)p Fw(\))295 b Fm(\000)g Fn(w)22545 45872 y Fj(b)22408 46208 y Fw(\011)23420 46407 y Fs(w)24173 46208 y Fw(\()p Fn(x)p Fw(\))p Fn(;)16669 b Fw(\(A.19\))9064 48502 y Fn(@)p 8694 49095 V 8694 50312 a(@)72 b(x)10426 49065 y Fj(b)10325 49401 y Fw(\010)11264 49600 y Fs(s)11755 49401 y Fw(\()p Fn(x)p Fw(\))1107 b(=)f Fn(w)17797 49065 y Fj(b)17697 49401 y Fw(\010)18636 49600 y Fs(s)19127 49401 y Fw(\()p Fn(x)p Fw(\))295 b Fm(\000)g Fw(Ai\()p Fn(x)p Fw(\))25809 47593 y Fj(Z)27138 47943 y Fs(s)26547 50600 y Fi(\0001)28497 49401 y Fw(d)p Fn(y)269 b Fw(Ai\()p Fn(y)48 b Fw(\))p Fn(e)33756 48852 y Fs(w)24 b(y)35302 49401 y Fm(\000)37069 48502 y Fn(@)p 36763 49095 1372 54 v 36763 50312 a(@)72 b(s)38368 49065 y Fj(b)38267 49401 y Fw(\010)39206 49600 y Fs(s)39697 49401 y Fw(\()p Fn(x)p Fw(\))p Fn(:)3432 53054 y Fw(In)433 b(the)g(end)g(only)h(four)g(terms)f(remain,)h(whic)-36 b(h)433 b(can)h(b)36 b(e)433 b(assem)-36 b(bled)433 b(as)3871 55691 y Fn(@)p 3564 56284 V 3564 57501 a(@)72 b(s)5115 56590 y Fw(~)-696 b Fn(g)47 b Fw(\()p Fn(s;)221 b(w)36 b Fw(\))369 b(=)10661 55115 y Fj(\020)11455 56590 y Fw(1)p Fm(\000)13138 54782 y Fj(Z)14467 55132 y Fi(1)13876 57789 y Fs(s)15684 56590 y Fw(d)p Fn(x)17283 56254 y Fj(b)17146 56590 y Fw(\011)18158 56789 y Fs(w)18912 56590 y Fw(\()p Fn(x)p Fw(\))p Fn(Q)p Fw(\()p Fn(x)p Fw(\))23444 55115 y Fj(\021\020)25252 54782 y(Z)26580 55132 y Fs(s)25990 57789 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Fn(e)23779 21925 y Fg(1)p 23779 22086 V 23779 22635 a(3)24318 22287 y Fs(w)25017 21975 y Fg(3)25828 22836 y Fm(\000)296 b(h)p Fn(e)28410 21925 y Fg(1)p 28409 22086 V 28409 22635 a(3)28948 22287 y Fs(w)29647 21975 y Fg(3)30300 22500 y Fj(b)30163 22836 y Fw(\011)31175 23035 y Fi(\000)p Fs(w)32661 22836 y Fn(;)221 b(P)34081 23035 y Fs(s)34573 22836 y Fn(Q)p Fm(i)6834 b Fw(\(A.24\))3432 25278 y(and)433 b(b)-36 b(y)433 b(\(D.3\))h(it)g(follo)-36 b(ws)435 b(that)f Fn(e)20343 24426 y Fg(1)p 20342 24587 V 20342 25135 a(3)20881 24788 y Fs(w)21580 24476 y Fg(3)22466 25278 y Fw(=)368 b Fm(h)p Fn( )25209 25477 y Fi(\000)p Fs(w)26695 25278 y Fn(;)221 b Fw(Ai)q Fm(i)p Fw(.)577 b(Moreo)-36 b(v)g(er,)8495 28888 y Fn(e)9231 27978 y Fg(1)p 9231 28139 V 9231 28687 a(3)9770 28340 y Fs(w)10469 28027 y Fg(3)11122 28553 y Fj(b)10985 28888 y Fw(\011)11997 29087 y Fi(\000)p Fs(w)13483 28888 y Fw(\()p Fn(x)p Fw(\))369 b(=)f Fn( )17829 29087 y Fi(\000)p Fs(w)19315 28888 y Fw(\()p Fn(x)p Fw(\))295 b Fm(\000)g Fn(e)23425 27978 y Fg(1)p 23425 28139 V 23425 28687 a(3)23964 28340 y Fs(w)24663 28027 y Fg(3)25401 27080 y Fj(Z)26729 27431 y Fi(1)26139 30088 y Fq(0)27947 28888 y Fw(d)p Fn(y)48 b(e)29955 28340 y Fi(\000)p Fs(w)24 b(y)32159 28888 y Fw(Ai\()p Fn(x)295 b Fw(+)g Fn(y)48 b Fw(\))p Fn(:)5063 b Fw(\(A.25\))3432 32807 y(An)1243 b(explicit)h(computation)g(sho)-36 b(ws)1244 b(that)f Fn(e)29079 31955 y Fg(1)p 29079 32116 V 29079 32664 a(3)29617 32317 y Fs(w)30316 32004 y Fg(3)31054 31737 y Fj(R)31940 32087 y Fi(1)31682 33268 y Fq(0)33158 32807 y Fw(d)p Fn(y)48 b(e)35166 32325 y Fi(\000)p Fs(w)24 b(y)37369 32807 y Fw(Ai)q(\()p Fn(x)846 b Fw(+)h Fn(y)48 b Fw(\))1747 b(=)3432 34412 y(\()p Fn(K)5045 34611 y Fq(Ai)6065 34412 y Fn( )6911 34611 y Fi(\000)p Fs(w)8397 34412 y Fw(\)\()p Fn(x)p Fw(\).)578 b(Th)-36 b(us)17260 36193 y Fn(e)17996 35283 y Fg(1)p 17996 35444 V 17996 35992 a(3)18535 35645 y Fs(w)19234 35332 y Fg(3)19887 35858 y Fj(b)19750 36193 y Fw(\011)20762 36392 y Fi(\000)p Fs(w)22617 36193 y Fw(=)369 b Fn( )24844 36392 y Fi(\000)p Fs(w)26625 36193 y Fm(\000)295 b Fn(K)29060 36392 y Fq(Ai)30081 36193 y Fn( )30927 36392 y Fi(\000)p Fs(w)32413 36193 y Fn(:)10180 b Fw(\(A.26\))3432 38636 y(Inserting)500 b(\(A.26\))h(in)g(\(A.24\))g (one)g(establishes)g(that)f(\(A.23\))i(equals)f Fn(e)39599 37784 y Fg(1)p 39599 37945 V 39599 38493 a(3)40138 38146 y Fs(w)40837 37833 y Fg(3)41353 38636 y Fn(a)p Fw(\()p Fn(s;)221 b Fm(\000)p Fn(w)36 b Fw(\).)p 45663 40241 45 878 v 45708 39407 781 45 v 45708 40241 V 46487 40241 45 878 v 3432 44678 a Fp(B)2151 b(Determinan)-60 b(tal)1274 b(\014elds:)2071 b(pro)60 b(of)1273 b(of)i(Prop)60 b(osi-)7113 47113 y(tions)717 b(3.2)g(and)f(3.3)3432 50421 y Fd(B.1)1792 b(No)598 b(b)50 b(oundary)599 b(sources)3432 52884 y Fw(T)-108 b(o)517 b(pro)-36 b(v)g(e)518 b(Prop)36 b(osition)518 b(3.3)g(w)-36 b(e)518 b(\014nd)e(it)i(computationally)g(con)-36 b(v)g(enien)g(t)517 b(to)g(appro)-36 b(x-)3432 54489 y(imate)531 b(the)f(exp)36 b(onen)-36 b(tial)531 b(distribution)f (through)g(a)h(geometric)h(one.)870 b(Then)530 b Fn(w)36 b Fw(\()p Fn(i;)221 b(j)75 b Fw(\),)3432 56095 y Fn(i;)221 b(j)444 b Fm(\025)369 b Fw(1,)344 b(are)322 b(indep)36 b(enden)-36 b(t)319 b(random)i(v)-72 b(ariables)322 b(with)g Fl(P)p Fw(\()p Fm(f)p Fn(w)36 b Fw(\()p Fn(i;)221 b(j)75 b Fw(\))366 b(=)j Fn(n)p Fm(g)p Fw(\))g(=)g(\(1)66 b Fm(\000)g Fn(q)48 b Fw(\))p Fn(q)45617 55613 y Fs(n)46242 56095 y Fw(,)3432 57700 y(0)700 b Fn(<)f(q)748 b(<)699 b Fw(1.)1162 b(The)628 b(RSK)g(construction,)676 b(as)628 b(explained)h(in)e(the)h(main)g(text,)677 b(can)3432 59305 y(b)36 b(e)546 b(carried)i(through)e(with)h(minor)g(mo)36 b(di\014cations,)576 b(compare)547 b(with)h([8].)920 b(The)547 b(lines)3432 60910 y Fn(j)443 b Fm(7!)370 b Fn(h)6860 61109 y Fs(`)7299 60910 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))431 b(tak)-36 b(e)430 b(v)-72 b(alues)430 b(in)f Fl(Z)h Fw(and)f(are)h(pinned)e(as)h Fn(h)31412 61109 y Fs(`)31852 60910 y Fw(\()p Fm(\006)p Fn(\034)76 b(;)221 b(\034)148 b Fw(\))370 b(=)e Fn(`)p Fw(,)430 b Fn(`)369 b Fw(=)g(0)p Fn(;)221 b Fm(\000)p Fw(1)p Fn(;)g(:)g(:)g(:)t Fw(.)3432 62515 y(The)501 b(w)-36 b(eigh)g(t)501 b(of)h(a)f(jump)g(of)h (size)g Fn(\016)551 b Fw(is)501 b(\()23576 61687 y Fm(p)p 24683 61687 625 54 v 828 x Fn(q)47 b Fw(\))25813 62033 y Fi(j)p Fs(\016)33 b Fi(j)26841 62515 y Fw(.)781 b(Let)500 b(us)h(denote)g(the)f(corresp)36 b(onding)3432 64120 y(p)g(oin)-36 b(t)433 b(random)g(\014eld)g(b)-36 b(y)433 b Fn(\036)16991 63638 y Fs(q)16991 64448 y(\034)17566 64120 y Fw(\()p Fn(j)r(;)221 b(n)p Fw(\),)436 b Fm(j)p Fn(j)75 b Fm(j)368 b(\024)h Fn(\034)148 b Fw(,)435 b Fn(n)369 b Fm(2)f Fl(Z)p Fw(,)435 b(i.e.,)9795 68216 y Fn(\036)10565 67667 y Fs(q)10565 68544 y(\034)11140 68216 y Fw(\()p Fn(j)r(;)221 b(n)p Fw(\))370 b(=)15800 65944 y Fj(\()16871 67306 y Fw(1)1301 b(if)434 b(there)f(is)h(an)f Fn(`)g Fw(suc)-36 b(h)433 b(that)g Fn(h)33932 67505 y Fs(`)34372 67306 y Fw(\()p Fn(j)r(;)221 b(\034)148 b Fw(\))370 b(=)e Fn(n;)16871 69232 y Fw(0)1301 b(otherwise)p Fn(:)43659 68216 y Fw(\(B.1\))24367 75163 y(41)p eop end end %%Page: 42 42 TeXDict begin HPSdict begin 42 41 bop 3432 7638 a Fw(It)433 b(is)h(determinan)-36 b(tal)433 b(and,)g(at)h(equal)g(times)g Fn(j)444 b Fw(=)368 b(2)p Fn(d)p Fw(,)434 b Fn(\034)518 b Fw(=)368 b(2)p Fn(m)p Fw(,)11207 11567 y Fl(E)12220 10092 y Fj(\020)13592 9906 y Fs(M)13242 10305 y Fj(Y)13233 13134 y Fs(k)24 b Fq(=1)15170 11567 y Fn(\036)15940 11018 y Fs(q)15940 11895 y(\034)16515 11567 y Fw(\(2)p Fn(d;)221 b(n)19705 11766 y Fs(k)20275 11567 y Fw(\))20781 10092 y Fj(\021)21943 11567 y Fw(=)369 b(det)25352 10491 y Fj(\000)25960 11567 y Fn(K)27162 10937 y Fs(q)27067 11937 y(m;d)28700 11567 y Fw(\()p Fn(n)29982 11766 y Fs(k)30551 11567 y Fn(;)221 b(n)31909 11766 y Fs(k)32423 11514 y Ff(0)32777 11567 y Fw(\))33283 10491 y Fj(\001)33892 12087 y Fq(1)p Fi(\024)p Fs(k)24 b(;k)36383 11835 y Ff(0)36681 12087 y Fi(\024)p Fs(M)38466 11567 y Fn(:)4832 b Fw(\(B.2\))3432 15549 y(W)-108 b(e)391 b(will)h(\014rst)e(compute)g(the)h(k)-36 b(ernel)391 b Fn(K)23100 14920 y Fs(q)23005 15919 y(m;d)25029 15549 y Fw(and)f(then)g(sho)-36 b(w)392 b(that)e(in)h(the)g(exp)36 b(onen)-36 b(tial)3432 17154 y(limit,)434 b(lattice)g(spacing)g Fn(")p Fw(,)g Fn(q)416 b Fw(=)369 b(1)295 b Fm(\000)g Fn(")p Fw(,)12845 19921 y(lim)12826 20718 y Fs(")p Fi(!)p Fq(0)14893 19921 y Fn(")15502 19372 y Fi(\000)p Fq(1)16759 19921 y Fn(K)17961 19371 y Fq(1)p Fi(\000)p Fs(")17866 20291 y(m;d)19653 18845 y Fj(\000)20262 19921 y Fm(b)p Fn(")21461 19372 y Fi(\000)p Fq(1)22719 19921 y Fn(x)p Fm(c)p Fn(;)221 b Fm(b)p Fn(")25829 19372 y Fi(\000)p Fq(1)27088 19921 y Fn(x)27827 19372 y Fi(0)28137 19921 y Fm(c)28727 18845 y Fj(\001)29706 19921 y Fw(=)368 b Fn(K)32193 20120 y Fs(m;d)33826 19921 y Fw(\()p Fn(x;)221 b(x)36392 19372 y Fi(0)36703 19921 y Fw(\))6450 b(\(B.3\))3432 23097 y(for)434 b Fn(x;)221 b(x)7479 22615 y Fi(0)8159 23097 y Fn(>)369 b Fw(0.)5383 24702 y(W)-108 b(e)315 b(ha)-36 b(v)g(e)316 b(to)g(compute)f Fn(K)18216 24072 y Fs(q)18121 25072 y(m;d)19753 24702 y Fw(,)340 b(for)316 b(whic)-36 b(h)315 b(w)-36 b(e)316 b(use)f(the)g(F)-108 b(ermion)315 b(formalism)i(of)f([18)q(].)3432 26307 y(Let)568 b([)p Fm(\000)p Fn(N)66 b(;)221 b Fm(\000)p Fn(N)527 b Fw(+)387 b(1)p Fn(;)221 b(:)g(:)g(:)j(;)d(N)139 b Fw(])600 b(=)f(\003)21201 26506 y Fs(N)22667 26307 y Fw(and)568 b Fm(F)701 b Fw(b)36 b(e)569 b(the)f(F)-108 b(ermionic)569 b(F)-108 b(o)36 b(c)-36 b(k)569 b(space)h(o)-36 b(v)g(er)3432 27912 y Fn(`)3978 28111 y Fq(2)4503 27912 y Fw(\(\003)5912 28111 y Fs(N)6808 27912 y Fw(\).)861 b(If)528 b Fn(A)g Fw(is)g(a)g(linear)g(op)36 b(erator)528 b(on)f Fn(`)25499 28111 y Fq(2)26025 27912 y Fw(\(\003)27434 28111 y Fs(N)28330 27912 y Fw(\),)552 b(then)526 b(\000\()p Fn(A)p Fw(\))h(denotes)g(its)h (second)3432 29517 y(quan)-36 b(tization)342 b(as)g(an)g(op)36 b(erator)342 b(on)f Fm(F)132 b Fw(.)548 b(Let)341 b Fn(P)26189 29035 y Fs(N)26008 29845 y Fi(\000)27427 29517 y Fw(b)36 b(e)342 b(the)f(pro)72 b(jection)343 b(on)-36 b(to)341 b([)p Fm(\000)p Fn(N)66 b(;)221 b(:)g(:)g(:)k(;)c Fw(0])3432 31122 y(and)385 b(let)h(\012)8683 31321 y Fs(N)9966 31122 y Fw(b)36 b(e)385 b(the)h(corresp)36 b(onding)385 b(Slater)h (determinan)-36 b(t.)561 b(W)-108 b(e)386 b(set,)396 b(as)386 b(op)36 b(erators)386 b(in)3432 32727 y Fn(`)3978 32926 y Fq(2)4503 32727 y Fw(,)16514 35421 y(\()p Fn(T)17963 34791 y Fs(q)17782 35711 y Fq(+)18569 35421 y Fw(\))19075 35620 y Fs(ij)20251 35421 y Fw(=)21632 33149 y Fj(\()22702 34511 y Fw(\()23208 33683 y Fm(p)p 24314 33683 625 54 v 24314 34511 a Fn(q)48 b Fw(\))25445 34029 y Fs(i)p Fi(\000)p Fs(j)28284 34511 y Fw(for)434 b Fn(i)369 b Fm(\025)g Fn(j)r(;)22702 36437 y Fw(0)4932 b(for)434 b Fn(i)369 b(<)g(j)r(;)43659 35421 y Fw(\(B.4\))3432 39075 y(and)483 b Fn(T)6954 38445 y Fs(q)6773 39365 y Fi(\000)8015 39075 y Fw(=)454 b(\()p Fn(T)10930 38445 y Fs(q)10749 39365 y Fq(+)11537 39075 y Fw(\))12043 38593 y Fi(\003)12568 39075 y Fw(.)730 b Fn(T)14602 38445 y Fs(q)32 b(;N)14421 39365 y Fq(+)16211 39075 y Fw(,)497 b Fn(T)18012 38445 y Fs(q)32 b(;N)17831 39365 y Fi(\000)20106 39075 y Fw(is)484 b(the)f(restriction)h(of)h Fn(T)32501 38445 y Fs(q)32320 39365 y Fq(+)33107 39075 y Fw(,)497 b Fn(T)34908 38445 y Fs(q)34727 39365 y Fi(\000)35998 39075 y Fw(to)484 b Fn(`)38184 39274 y Fq(2)38710 39075 y Fw(\(\003)40119 39274 y Fs(N)41015 39075 y Fw(\).)729 b(Finally)3432 40680 y(let)433 b Fn(a)p Fw(\()p Fn(j)75 b Fw(\))433 b(b)36 b(e)433 b(the)g(F)-108 b(ermion)433 b(\014eld)h(with)f(index)h Fn(j)75 b Fw(,)433 b Fm(j)p Fn(j)75 b Fm(j)369 b(\024)g Fn(N)139 b Fw(.)578 b(Then)4378 44038 y Fn(K)5580 43409 y Fs(q)5485 44408 y(m;d)7118 44038 y Fw(\()p Fn(i;)221 b(j)75 b Fw(\))369 b(=)827 b(lim)11518 44865 y Fs(N)94 b Fi(!1)15164 43140 y Fw(1)p 14596 43733 1787 54 v 14596 44950 a Fn(Z)15486 45149 y Fs(N)16515 44038 y Fm(h)p Fw(\012)17971 44237 y Fs(N)18868 44038 y Fn(;)221 b Fw(\000\()p Fn(T)21712 43409 y Fs(q)32 b(;N)21531 44328 y Fi(\000)23322 44038 y Fn(T)24265 43409 y Fs(q)g(;N)24084 44328 y Fq(+)25874 44038 y Fw(\))26380 43490 y Fs(m)p Fq(+)p Fs(d)28484 44038 y Fn(a)29167 43490 y Fi(\003)29692 44038 y Fw(\()p Fn(j)75 b Fw(\))p Fn(a)p Fw(\()p Fn(i)p Fw(\)\000\()p Fn(T)35718 43409 y Fs(q)32 b(;N)35537 44328 y Fi(\000)37326 44038 y Fn(T)38269 43409 y Fs(q)g(;N)38088 44328 y Fq(+)39879 44038 y Fw(\))40385 43490 y Fs(m)p Fi(\000)p Fs(d)42488 44038 y Fw(\012)43427 44237 y Fs(N)44324 44038 y Fm(i)44841 44237 y Fi(F)43659 46256 y Fw(\(B.5\))3432 47861 y(with)433 b(the)g(normalization)15873 50627 y Fn(Z)16763 50826 y Fs(N)18029 50627 y Fw(=)368 b Fm(h)p Fw(\012)20865 50826 y Fs(N)21762 50627 y Fn(;)221 b Fw(\000\()p Fn(T)24606 49997 y Fs(q)32 b(;N)24425 50917 y Fi(\000)26216 50627 y Fn(T)27159 49997 y Fs(q)g(;N)26978 50917 y Fq(+)28768 50627 y Fw(\))29274 50079 y Fq(2)p Fs(m)30632 50627 y Fw(\012)31571 50826 y Fs(N)32468 50627 y Fm(i)32985 50826 y Fi(F)33800 50627 y Fn(:)9498 b Fw(\(B.6\))3432 53394 y(Here)428 b Fm(h\001)p Fn(;)221 b Fm(\001i)8851 53593 y Fi(F)10094 53394 y Fw(denotes)428 b(the)g(inner)f(pro)36 b(duct)427 b(in)h(F)-108 b(o)36 b(c)-36 b(k)429 b(space.)577 b(W)-108 b(orking)429 b(out)e(the)h(limit,)3432 54999 y(see)433 b([18)q(],)h(results)g(in)6254 57765 y Fn(K)7456 57135 y Fs(q)7361 58135 y(m;d)8994 57765 y Fw(\()p Fn(i;)221 b(j)75 b Fw(\))369 b(=)13394 56689 y Fj(\000)14003 57765 y Fw(\()p Fn(T)15452 57135 y Fs(q)15271 58055 y Fq(+)16059 57765 y Fw(\))16565 57216 y Fs(m)p Fq(+)p Fs(d)18668 57765 y Fw(\()p Fn(T)20117 57135 y Fs(q)19936 58055 y Fi(\000)20723 57765 y Fw(\))21229 57216 y Fi(\000)p Fq(\()p Fs(m)p Fi(\000)p Fs(d)p Fq(\))24796 57765 y Fn(P)25634 57964 y Fi(\000)26422 57765 y Fw(\()p Fn(T)27871 57135 y Fs(q)27690 58055 y Fq(+)28477 57765 y Fw(\))28983 57216 y Fi(\000)p Fq(\()p Fs(m)p Fq(+)p Fs(d)p Fq(\))32550 57765 y Fw(\()p Fn(T)33999 57135 y Fs(q)33818 58055 y Fi(\000)34606 57765 y Fw(\))35112 57216 y Fs(m)p Fi(\000)p Fs(d)37215 56689 y Fj(\001)37824 57765 y Fw(\()p Fn(i;)221 b(j)75 b Fw(\))p Fn(;)2823 b Fw(\(B.7\))3432 60666 y(where)433 b Fn(P)8027 60865 y Fi(\000)9184 60666 y Fw(=)368 b(lim)12371 60865 y Fs(N)94 b Fi(!1)15370 60666 y Fn(P)16389 60184 y Fs(N)16208 60995 y Fi(\000)17719 60666 y Fw(pro)72 b(jects)434 b(on)-36 b(to)434 b Fl(Z)26523 60865 y Fi(\000)27680 60666 y Fw(=)369 b(\()p Fn(:)221 b(:)g(:)h(;)f Fm(\000)p Fw(1)p Fn(;)g Fw(0].)5383 62456 y(In)443 b(F)-108 b(ourier)443 b(space)g Fn(T)15954 61826 y Fs(q)15773 62746 y Fq(+)17004 62456 y Fw(is)h(m)-36 b(ultiplication)444 b(b)-36 b(y)443 b(\(1)302 b Fm(\000)31287 61628 y(p)p 32394 61628 625 54 v 828 x Fn(q)47 b(e)33621 61974 y Fi(\000)p Fs(ik)35242 62456 y Fw(\))35748 61974 y Fi(\000)p Fq(1)37392 62456 y Fw(=)39044 62120 y(^)38789 62456 y Fn(T)39732 61826 y Fs(q)39551 62746 y Fq(+)40338 62456 y Fw(\()p Fn(k)e Fw(\))444 b(and)f Fn(T)45997 61826 y Fs(q)45816 62746 y Fi(\000)3432 64272 y Fw(m)-36 b(ultiplication)430 b(b)-36 b(y)429 b(\(1)288 b Fm(\000)16340 63445 y(p)p 17447 63445 V 827 x Fn(q)47 b(e)18674 63790 y Fs(ik)19564 64272 y Fw(\))20070 63790 y Fi(\000)p Fq(1)21696 64272 y Fw(=)23331 63937 y(^)23077 64272 y Fn(T)24020 63643 y Fs(q)23839 64562 y Fi(\000)24626 64272 y Fw(\()p Fn(k)e Fw(\).)577 b(The)430 b(rescaling)g(in)g(\(B.3\))g(amoun)-36 b(ts)429 b(to)3432 65877 y(replacing)434 b Fn(q)480 b Fw(b)-36 b(y)434 b(1)295 b Fm(\000)h Fn(")433 b Fw(and)g Fn(k)479 b Fw(b)-36 b(y)433 b Fn("k)45 b Fw(.)578 b(Then)6607 68644 y(lim)6588 69441 y Fs(")p Fi(!)p Fq(0)8655 68644 y Fn(")9518 68308 y Fw(^)9264 68644 y Fn(T)10207 68095 y Fq(1)p Fi(\000)p Fs(")10026 68972 y Fq(+)11899 68644 y Fw(\()p Fn("k)45 b Fw(\))368 b(=)h(\(1)295 b(+)18881 68121 y Fq(1)p 18881 68338 471 54 v 18881 69102 a(2)19484 68644 y Fn(ik)45 b Fw(\))21155 68095 y Fi(\000)p Fq(1)22412 68644 y Fn(;)1542 b Fw(lim)24295 69441 y Fs(")p Fi(!)p Fq(0)26363 68644 y Fn(")27226 68308 y Fw(^)26972 68644 y Fn(T)27915 68095 y Fq(1)p Fi(\000)p Fs(")27734 68972 y Fi(\000)29606 68644 y Fw(\()p Fn("k)45 b Fw(\))369 b(=)f(\(1)296 b Fm(\000)36610 68121 y Fq(1)p 36610 68338 V 36610 69102 a(2)37213 68644 y Fn(ik)45 b Fw(\))38884 68095 y Fi(\000)p Fq(1)40141 68644 y Fn(:)3157 b Fw(\(B.8\))3432 71842 y(By)434 b(inserting)f(the)g(limit)h(\(B.8\))g(in)g(\(B.7\),)g (the)f(claim)h(\(B.3\))g(follo)-36 b(ws.)24367 75163 y(42)p eop end end %%Page: 43 43 TeXDict begin HPSdict begin 43 42 bop 3432 7638 a Fd(B.2)1792 b(Boundary)599 b(sources)3432 10101 y Fw(W)-108 b(e)647 b(add)f(b)36 b(oundary)647 b(sources)g(through)f(the)g(random)h(v)-72 b(ariables)648 b Fn(w)36 b Fw(\(0)p Fn(;)221 b Fw(0\),)701 b Fn(w)36 b Fw(\()p Fn(j)r(;)221 b Fw(0\),)3432 11706 y Fn(w)36 b Fw(\(0)p Fn(;)221 b(j)75 b Fw(\),)332 b Fn(j)444 b Fm(\025)369 b Fw(1.)536 b(They)307 b(are)g(indep)36 b(enden)-36 b(t)304 b(and)i(geometrically)j(distributed)c(according) 3432 13311 y(to)13216 16245 y Fl(P)p Fw(\()p Fm(f)p Fn(w)36 b Fw(\(0)p Fn(;)221 b Fw(0\))367 b(=)i Fn(n)p Fm(g)p Fw(\))1107 b(=)g(\(1)295 b Fm(\000)g Fn(\013)8 b(\014)74 b Fw(\)\()p Fn(\013)8 b(\014)74 b 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b(Let)15018 35026 y Fn(f)15659 35225 y Fs(\013)16319 35026 y Fw(\()p Fn(j)75 b Fw(\))368 b(=)h Fn(\013)20528 34477 y Fs(j)21016 35026 y Fn(;)1522 b(a)p Fw(\()p Fn(f)24729 35225 y Fs(\013)25388 35026 y Fw(\))369 b(=)27644 33764 y Fj(X)27742 36583 y Fs(j)51 b Fi(2)p Fk(Z)29784 35026 y Fn(f)30425 35225 y Fs(\013)31085 35026 y Fw(\()p Fn(j)75 b Fw(\))p Fn(a)p Fw(\()p Fn(j)g Fw(\))7990 b(\(B.10\))3432 39412 y(and)415 b(let)h(\012)8743 38862 y Fi(\000)8743 39772 y Fs(N)10055 39412 y Fw(b)36 b(e)416 b(the)f(ground)g(state)h(v) -36 b(ector)416 b(with)g(sites)g Fm(\000)p Fn(N)66 b(;)221 b(:)g(:)g(:)j(;)d Fm(\000)p Fw(1)417 b(\014lled)e(and)h(sites)3432 41017 y(0)p Fn(;)221 b(:)g(:)g(:)j(;)d(N)569 b Fw(empt)-36 b(y)-108 b(.)577 b(Then)431 b Fn(a)17054 40535 y Fi(\003)17579 41017 y Fw(\()p Fn(f)18726 41216 y Fs(\013)19386 41017 y Fw(\)\012)20831 40467 y Fi(\000)20831 41377 y Fs(N)22158 41017 y Fw(giv)-36 b(es)432 b(the)e(correct)h(w)-36 b(eigh)g(t)431 b(to)g(the)f(jump)h(at)g(the)3432 42622 y(righ)-36 b(t)329 b(b)36 b(oundary)-108 b(,)350 b(corresp)36 b(ondingly)331 b(for)g(the)e(left)i(b)36 b(oundary)-108 b(.)543 b(Therefore)330 b(the)g(de\014ning)3432 44227 y(k)-36 b(ernel)433 b(is)h(giv)-36 b(en)434 b(through)7576 47679 y Fn(K)8778 47000 y Fq(\(2)p Fs(m)p Fq(+1\))8683 48049 y Fs(q)32 b(;\013\014)12071 47679 y Fw(\(2)p Fn(d;)221 b(i)p Fw(;)g(2)p Fn(d;)g(j)75 b Fw(\))1108 b(=)1565 b(lim)21765 48505 y Fs(N)94 b Fi(!1)25410 46780 y Fw(1)p 24842 47373 1787 54 v 24842 48590 a Fn(Z)25732 48789 y Fs(N)26761 47679 y Fm(h)p Fw(\012)28217 47129 y Fi(\000)28217 48039 y Fs(N)29114 47679 y Fn(;)221 b(a)p Fw(\()p Fn(f)31526 47878 y Fs(\014)32155 47679 y Fw(\)\000\()p Fn(T)34923 47049 y Fs(q)32 b(;N)34742 47969 y Fi(\000)36532 47679 y Fn(T)37475 47049 y Fs(q)g(;N)37294 47969 y Fq(+)39085 47679 y Fw(\))39591 47130 y Fs(m)p Fq(+)p Fs(d)43008 47679 y Fw(\(B.11\))21765 50467 y Fn(a)22448 49919 y Fi(\003)22973 50467 y Fw(\()p Fn(i)p Fw(\))p Fn(a)p Fw(\()p Fn(j)75 b Fw(\)\000\()p Fn(T)28999 49837 y Fs(q)32 b(;N)28818 50757 y Fi(\000)30607 50467 y Fn(T)31550 49837 y Fs(q)g(;N)31369 50757 y Fq(+)33160 50467 y Fw(\))33666 49919 y Fs(m)p Fi(\000)p Fs(d)35769 50467 y Fn(a)36452 49919 y Fi(\003)36978 50467 y Fw(\()p Fn(f)38125 50666 y Fs(\013)38784 50467 y Fw(\)\012)40229 49917 y Fi(\000)40229 50827 y Fs(N)41126 50467 y Fm(i)41643 50666 y Fi(F)3432 53400 y Fw(with)433 b(the)g(normalization)12630 56334 y Fn(Z)13520 56533 y Fs(N)14786 56334 y Fw(=)369 b Fm(h)p Fw(\012)17623 55784 y Fi(\000)17623 56694 y Fs(N)18519 56334 y Fn(;)221 b(a)p Fw(\()p Fn(f)20931 56533 y Fs(\014)21560 56334 y Fw(\)\000\()p Fn(T)24328 55704 y Fs(q)32 b(;N)24147 56624 y Fi(\000)25937 56334 y Fn(T)26880 55704 y Fs(q)g(;N)26699 56624 y Fq(+)28490 56334 y Fw(\))28996 55785 y Fq(2)p Fs(m)30354 56334 y Fn(a)31037 55785 y Fi(\003)31563 56334 y Fw(\()p Fn(f)32710 56533 y Fs(\013)33369 56334 y Fw(\)\012)34814 55784 y Fi(\000)34814 56694 y Fs(N)35711 56334 y Fm(i)36228 56533 y Fi(F)37043 56334 y Fn(:)5604 b Fw(\(B.12\))3432 59267 y(W)-108 b(e)433 b(note)h(that)17852 60872 y(\000\()p Fn(A)p Fw(\))p Fn(a)21335 60324 y Fi(\003)21860 60872 y Fw(\()p Fn(f)142 b Fw(\))369 b(=)g Fn(a)26088 60324 y Fi(\003)26613 60872 y Fw(\()p Fn(Af)142 b Fw(\)\000\()p Fn(A)p Fw(\))10825 b(\(B.13\))3432 63197 y(and)13754 66130 y Fn(T)14697 65501 y Fs(q)14516 66420 y Fq(+)15303 66130 y Fn(f)15944 66329 y Fs(\013)17711 66130 y Fw(=)1107 b(\(1)295 b Fm(\000)g Fn(\013)23444 65582 y Fi(\000)p Fq(1)24703 65238 y Fm(p)p 25810 65238 625 54 v 892 x Fn(q)47 b Fw(\))26940 65582 y Fi(\000)p Fq(1)28197 66130 y Fn(f)28838 66329 y Fs(\013)29498 66130 y Fn(;)31382 65238 y Fm(p)p 32489 65238 V 892 x Fn(q)416 b(<)368 b(\013)8 b(;)13785 68711 y(T)14728 68081 y Fs(q)14547 69001 y Fi(\000)15334 68711 y Fn(f)15975 68910 y Fs(\014)17711 68711 y Fw(=)1107 b(\(1)295 b Fm(\000)g Fn(\014)23417 67818 y Fm(p)p 24524 67818 V 893 x Fn(q)47 b Fw(\))25654 68162 y Fi(\000)p Fq(1)26912 68711 y Fn(f)27553 68910 y Fs(\014)28182 68711 y Fn(;)1522 b(\014)443 b(<)33296 67812 y Fw(1)p 32756 68405 1732 54 v 32756 68794 a Fm(p)p 33863 68794 625 54 v 828 x Fn(q)34620 68711 y(:)8027 b Fw(\(B.14\))24367 75163 y(43)p eop end end %%Page: 44 44 TeXDict begin HPSdict begin 44 43 bop 3432 7638 a Fw(Therefore)6094 10809 y Fn(K)7296 10130 y Fq(\(2)p Fs(m)p Fq(+1\))7201 11179 y Fs(q)32 b(;\013\014)10588 10809 y Fw(\(2)p Fn(d;)221 b(i)p Fw(;)g(2)p Fn(d;)g(j)75 b Fw(\))1109 b(=)1565 b(lim)20283 11636 y Fs(N)94 b Fi(!1)24621 9911 y Fw(1)p 23360 10504 3174 54 v 23636 11545 a(~)23360 11881 y Fn(Z)24250 12080 y Fs(N)42 b(;\013\014)26666 10809 y Fm(h)p Fw(\012)28122 10259 y Fi(\000)28122 11169 y Fs(N)29019 10809 y Fn(;)221 b Fw(\000\()p Fn(T)31863 10180 y Fs(q)32 b(;N)31682 11099 y Fi(\000)33473 10809 y Fn(T)34416 10180 y Fs(q)g(;N)34235 11099 y Fq(+)36025 10809 y Fw(\))36531 10261 y Fs(m)p Fq(+)p Fs(d)43008 10809 y Fw(\(B.15\))20283 13930 y Fn(a)p Fw(\()p Fn(f)22113 14129 y Fs(\014)22741 13930 y Fw(\))p Fn(a)23930 13381 y Fi(\003)24455 13930 y Fw(\()p Fn(i)p Fw(\))p Fn(a)p Fw(\()p Fn(j)75 b Fw(\))p Fn(a)28902 13381 y Fi(\003)29427 13930 y Fw(\()p Fn(f)30574 14129 y Fs(\013)31233 13930 y Fw(\)\000\()p Fn(T)34001 13300 y Fs(q)32 b(;N)33820 14220 y Fi(\000)35610 13930 y Fn(T)36553 13300 y Fs(q)g(;N)36372 14220 y Fq(+)38163 13930 y Fw(\))38669 13381 y Fs(m)p Fi(\000)p Fs(d)40772 13930 y Fw(\012)41711 13380 y Fi(\000)41711 14290 y Fs(N)42608 13930 y Fm(i)43125 14129 y Fi(F)3432 16841 y Fw(with)5526 19416 y(~)5250 19752 y Fn(Z)6140 19951 y Fs(N)42 b(;\013\014)8793 19752 y Fw(=)368 b Fm(h)p Fw(\012)11629 19202 y Fi(\000)11629 20112 y Fs(N)12526 19752 y Fn(;)221 b Fw(\000\()p Fn(T)15370 19122 y Fs(q)32 b(;N)15189 20042 y Fi(\000)16980 19752 y Fn(T)17923 19122 y Fs(q)g(;N)17742 20042 y Fq(+)19532 19752 y Fw(\))20038 19203 y Fs(m)p Fq(+)p Fs(d)22142 19752 y Fn(a)p Fw(\()p Fn(f)23972 19951 y Fs(\014)24600 19752 y Fw(\))p Fn(a)25789 19203 y Fi(\003)26315 19752 y Fw(\()p Fn(f)27462 19951 y Fs(\013)28121 19752 y Fw(\)\000\()p Fn(T)30889 19122 y Fs(q)g(;N)30708 20042 y Fi(\000)32498 19752 y Fn(T)33441 19122 y Fs(q)g(;N)33260 20042 y Fq(+)35051 19752 y Fw(\))35557 19203 y Fs(m)p Fi(\000)p Fs(d)37660 19752 y Fw(\012)38599 19202 y Fi(\000)38599 20112 y Fs(N)39496 19752 y Fm(i)40013 19951 y Fi(F)40829 19752 y Fn(:)1818 b Fw(\(B.16\))3432 22663 y(F)-108 b(ollo)-36 b(wing)434 b(the)f(steps)g(of)i(Section)e (B.1,)i(one)e(concludes)g(that)6685 25574 y(lim)6226 26400 y Fs(N)94 b Fi(!1)9447 25238 y Fw(~)9171 25574 y Fn(Z)10061 25773 y Fs(N)42 b(;\013\014)12713 25574 y Fw(=)14370 25238 y(~)14094 25574 y Fn(Z)14984 25773 y Fs(\013\014)16586 25574 y Fw(=)369 b(\()p Fn(\013)8 b(\014)74 b Fw(\))20622 25025 y Fi(\000)p Fq(1)21880 25574 y Fm(h)p Fn(f)23038 25773 y Fs(\013)23697 25574 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)d Fn(K)28388 24944 y Fs(q)28293 25943 y(m;d)29925 25574 y Fw(\))p Fn(f)31072 25773 y Fs(\014)31701 25574 y Fm(i)32218 25773 y Fs(`)32602 25896 y Fg(2)43008 25574 y Fw(\(B.17\))6226 29202 y(=)369 b(\(1)295 b Fm(\000)h 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Fw(A)-36 b(t)506 b(this)h(stage)h(it)f(is)h(of)g(use)f(to)g (recall)h(a)f(general)h(prop)36 b(ert)-36 b(y)507 b(of)h(quasifree)g (states)3432 34505 y(on)543 b(CAR-algebras.)909 b(Let)543 b Fm(A)g Fw(b)36 b(e)543 b(a)g(CAR)h(algebra)h(indexed)e(b)-36 b(y)543 b Fl(Z)h Fw(and)f(let)h Fn(!)591 b Fw(b)36 b(e)543 b(a)3432 36110 y(quasifree)434 b(linear)g(functional)g(\(a)g(state\))f (on)h Fm(A)p Fw(,)f(uniquely)h(de\014ned)e(through)11256 39021 y Fn(!)48 b Fw(\()p Fn(a)p Fw(\()p Fn(j)75 b Fw(\)\))367 b(=)i Fn(!)48 b Fw(\()p Fn(a)19229 38472 y Fi(\003)19754 39021 y Fw(\()p Fn(j)75 b Fw(\)\))368 b(=)g(0)p Fn(;)1523 b(!)48 b Fw(\()p Fn(a)28214 38472 y Fi(\003)28739 39021 y Fw(\()p Fn(j)75 b Fw(\))p Fn(a)p Fw(\()p Fn(j)g Fw(\)\))368 b(=)g Fn(K)95 b Fw(\()p Fn(i;)221 b(j)75 b Fw(\))4229 b(\(B.18\))3432 41932 y(with)505 b Fn(K)600 b Fw(a)506 b(p)36 b(ositiv)-36 b(e)506 b(linear)g(op)36 b(erator)505 b(on)g Fn(`)25632 42131 y Fq(2)26158 41932 y Fw(,)523 b Fm(k)p Fn(K)95 b Fm(k)492 b(\024)f Fw(1.)794 b(Let)505 b Fn(f)633 b Fm(2)490 b Fn(`)38991 42131 y Fq(2)40022 41932 y Fw(and)505 b Fn(a)p Fw(\()p Fn(f)142 b Fw(\))490 b(=)3432 42541 y Fj(P)4834 43924 y Fs(j)51 b Fi(2)p Fk(Z)6833 43537 y Fn(f)142 b Fw(\()p Fn(j)75 b Fw(\))p Fn(a)p Fw(\()p Fn(j)g Fw(\).)577 b(F)-108 b(or)434 b Fn(f)15459 43736 y Fs(`)15898 43537 y Fn(;)221 b(f)17121 43736 y Fs(r)17997 43537 y Fm(2)369 b Fn(`)19798 43736 y Fq(2)20757 43537 y Fw(w)-36 b(e)434 b(de\014ne)12551 46448 y Fn(Z)464 b Fw(=)369 b Fn(!)48 b Fw(\()p Fn(a)p Fw(\()p Fn(f)18480 46647 y Fs(`)18919 46448 y Fw(\))p Fn(a)20108 45899 y Fi(\003)20633 46448 y Fw(\()p Fn(f)21780 46647 y Fs(r)22286 46448 y Fw(\)\))369 b(=)f Fm(h)p Fn(f)26205 46647 y Fs(r)26711 46448 y Fn(;)221 b Fw(\()p Fl(1)299 b Fm(\000)d Fn(K)95 b Fw(\))p Fn(f)32549 46647 y Fs(`)32988 46448 y Fm(i)33505 46647 y Fs(`)33889 46770 y Fg(2)34775 46448 y Fn(<)368 b Fm(1)5525 b Fw(\(B.19\))3432 49359 y(and)433 b(a)h(linear)f (functional)538 b(~)-754 b Fn(!)482 b Fw(through)14753 52788 y(~)-754 b Fn(!)48 b Fw(\()p Fn(A)p Fw(\))369 b(=)19544 51890 y(1)p 19376 52483 986 54 v 19376 53700 a Fn(Z)20495 52788 y(!)48 b Fw(\()p Fn(a)p Fw(\()p Fn(f)23689 52987 y Fs(`)24127 52788 y Fw(\))p Fn(Aa)26291 52240 y Fi(\003)26817 52788 y Fw(\()p Fn(f)27964 52987 y Fs(r)28470 52788 y Fw(\)\))p Fn(;)1522 b(A)369 b Fm(2)f(A)p Fn(:)7623 b Fw(\(B.20\))3432 56112 y(Then)628 b(~)-754 b Fn(!)48 b Fw(\()p Fl(1)p Fw(\))527 b(=)d(1)h(and)628 b(~)-754 b Fn(!)573 b Fw(is)525 b(again)g(quasifree)h(with)629 b(~)-754 b Fn(!)48 b Fw(\()p Fn(a)p Fw(\()p Fn(j)75 b Fw(\)\))523 b(=)628 b(~)-754 b Fn(!)47 b Fw(\()p Fn(a)38616 55630 y Fi(\003)39142 56112 y Fw(\()p Fn(j)75 b Fw(\)\))523 b(=)h(0)i(and)3432 57718 y(co)-36 b(v)-72 b(ariance)7965 60889 y(~)-754 b Fn(!)48 b Fw(\()p Fn(a)9908 60340 y Fi(\003)10433 60889 y Fw(\()p Fn(i)p Fw(\))p Fn(a)p Fw(\()p Fn(j)75 b Fw(\)\))367 b(=)i Fn(K)95 b Fw(\()p Fn(i;)221 b(j)75 b Fw(\))295 b(+)22206 59990 y(1)p 22039 60583 V 22039 61800 a Fn(Z)23157 60889 y Fw(\()p Fl(1)j 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v 828 x Fn(q)416 b(<)369 b(\013)8 b(;)221 b(\014)444 b(<)32443 71014 y Fm(p)p 33550 71014 V 828 x Fn(q)34174 71258 y Fi(\000)p Fq(1)35432 71842 y Fw(.)24367 75163 y(44)p eop end end %%Page: 45 45 TeXDict begin HPSdict begin 45 44 bop 5383 7638 a Fw(W)-108 b(e)433 b(tak)-36 b(e)434 b(the)f(exp)36 b(onen)-36 b(tial)434 b(limit)g(b)-36 b(y)434 b(c)-36 b(ho)36 b(osing)4096 11090 y Fn(q)416 b Fw(=)369 b(1)237 b Fm(\000)g Fn(";)1522 b(\013)378 b Fw(=)368 b(1)237 b Fm(\000)g Fn("a;)1522 b(\014)443 b Fw(=)369 b(1)237 b Fm(\000)g Fn("b;)1522 b Fm(\000)27962 10191 y Fw(1)p 27962 10785 651 54 v 27962 12001 a(2)29114 11090 y Fn(<)369 b(a;)221 b(b)369 b(<)34195 10191 y Fw(1)p 34195 10785 V 34195 12001 a(2)34979 11090 y Fn(;)1522 b(a)237 b Fw(+)f Fn(b)369 b(>)g Fw(0)p Fn(:)664 b Fw(\(B.23\))3432 14363 y(Then)15957 15968 y(lim)15937 16765 y Fs(")p Fi(!)p Fq(0)18004 15968 y Fn(f)18645 16167 y Fq(1)p Fi(\000)p Fs("a)20837 15968 y Fw(\()p Fm(b)p Fn(x=")p Fm(c)p Fw(\))370 b(=)e Fn( )27623 16167 y Fs(a)28178 15968 y Fw(\()p Fn(x)p Fw(\))h(=)f Fn(e)32281 15419 y Fi(\000)p Fs(ax)43008 15968 y Fw(\(B.24\))3432 18650 y(and,)433 b(for)h Fn(x;)221 b(y)418 b(>)368 b Fw(0,)11099 21584 y(lim)11080 22381 y Fs(")p Fi(!)p Fq(0)13147 21584 y Fn(")13756 21035 y Fi(\000)p Fq(1)15013 21584 y Fn(K)16215 20905 y Fq(\(2)p Fs(m)p Fq(+1\))16120 21954 y(1)p Fi(\000)p Fs(";\013\014)19507 20508 y Fj(\000)20116 21584 y Fw(2)p Fn(d;)221 b Fm(b)p Fn(")23223 21035 y Fi(\000)p Fq(1)24482 21584 y Fn(x)p Fm(c)p Fw(;)g(2)p Fn(d;)g Fm(b)p Fn(")29500 21035 y Fi(\000)p Fq(1)30760 21584 y Fn(y)48 b Fm(c)32032 20508 y Fj(\001)43008 21584 y Fw(\(B.25\))11080 24711 y(=)368 b Fn(K)13567 24910 y Fs(m;d)15200 24711 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))295 b(+)20680 23812 y(1)p 19950 24406 2110 54 v 19950 25622 a Fn(Z)20840 25821 y Fs(a;b)22192 24711 y Fw(\()p Fl(1)k Fm(\000)c Fn(K)26205 24910 y Fs(m;d)27838 24711 y Fw(\))p Fn( )29190 24910 y Fs(b)29647 24711 y Fw(\()p Fn(x)p Fw(\)\()p Fl(1)j Fm(\000)d Fn(K)35410 24910 y Fs(m;d)37043 24711 y Fw(\))37549 24163 y Fi(\003)38075 24711 y Fn( )38921 24910 y Fs(a)39475 24711 y Fw(\()p Fn(y)48 b Fw(\))3432 28429 y(with)433 b Fn(Z)7284 28628 y Fs(a;b)8937 28429 y Fw(giv)-36 b(en)434 b(in)g(\(3.24\).)579 b(This)434 b(completes)g(the)f(pro)36 b(of)434 b(of)g(Prop)36 b(osition)434 b(3.3.)3432 32280 y Fd(B.3)1792 b(Pro)50 b(of)598 b(of)f(Prop)50 b(osition)599 b(3.2)3432 34550 y Fo(Pr)-66 b(o)g(of)480 b(of)i(Pr)-66 b(op)g(osition)480 b(3.2.)632 b Fw(By)452 b(Prop)36 b(osition)453 b(C.1)f Fn(K)30749 34749 y Fs(m;d)32834 34550 y Fw(is)g(a)g(similarit)-36 b(y)453 b(transform)3432 36155 y(of)436 b(the)f(Laguerre)h(k)-36 b(ernel.)585 b(Therefore)437 b Fn(K)23938 36354 y Fs(m;d)25943 36155 y Fw(=)373 b(\()p Fn(K)28941 36354 y Fs(m;d)30573 36155 y Fw(\))31079 35673 y Fq(2)31605 36155 y Fw(,)437 b Fm(k)p Fn(K)34174 36354 y Fs(m;d)35807 36155 y Fm(k)373 b Fw(=)f(1,)437 b Fn(P)40514 36354 y Fs(u)41115 36155 y Fn(K)42222 36354 y Fs(m;d)43854 36155 y Fn(P)44692 36354 y Fs(u)45729 36155 y Fw(is)3432 37760 y(trace)495 b(class,)513 b(and)495 b(all)i(eigen)-36 b(v)-72 b(alues)496 b(of)h Fn(P)23865 37959 y Fs(u)24465 37760 y Fn(K)25572 37959 y Fs(m;d)27205 37760 y Fn(P)28043 37959 y Fs(u)29139 37760 y Fw(are)f(in)g(the)f(in)-36 b(terv)-72 b(al)496 b([0)p Fn(;)221 b Fw(1].)767 b(Th)-36 b(us)3432 39365 y(w)g(e)503 b(only)i(pro)-36 b(v)g(e)503 b(that,)521 b(if)504 b Fn(u)488 b(>)g Fw(0,)521 b(1)504 b(is)g(not)f(in)h(the)f(sp) 36 b(ectrum)502 b(of)i Fn(P)37654 39564 y Fs(u)38255 39365 y Fn(K)39362 39564 y Fs(m;d)40994 39365 y Fn(P)41832 39564 y Fs(u)42433 39365 y Fw(,)521 b(whic)-36 b(h)3432 40970 y(is)531 b(accomplished)f(b)-36 b(y)531 b Fo(r)-66 b(e)g(ductio)553 b(ad)h(absur)-66 b(dum)p Fw(.)870 b(Assume)530 b(that)h Fn( )581 b Fm(2)534 b Fn(L)39887 40488 y Fq(2)40412 40970 y Fw(\()p Fl(R)41998 41169 y Fq(+)42789 40970 y Fw(\))c(is)h(an)3432 42575 y(eigenfunction)433 b(for)h(eigen)-36 b(v)-72 b(alue)435 b(1,)20260 45509 y Fn(P)21098 45708 y Fs(u)21699 45509 y Fn(K)22806 45708 y Fs(m;d)24438 45509 y Fn(P)25276 45708 y Fs(u)25877 45509 y Fn( )416 b Fw(=)369 b Fn( )48 b(:)13233 b Fw(\(B.26\))3432 48442 y(Then)433 b Fn( )48 b Fw(\()p Fn(x)p Fw(\))368 b(=)g(0)434 b(for)g Fn(x)369 b Fm(2)g Fw([0)p Fn(;)221 b(u)p Fw(\).)580 b(On)433 b(the)g(other)g(hand,)17644 51375 y Fm(k)p Fn( )48 b Fm(k)369 b(\024)g(k)p Fn(K)23408 51574 y Fs(m;d)25041 51375 y Fn(P)25879 51574 y Fs(u)26479 51375 y Fn( )48 b Fm(k)369 b(\024)g(k)p Fn( )48 b Fm(k)p Fn(:)10617 b Fw(\(B.27\))3432 54309 y(Hence)433 b Fm(k)p Fn(K)9068 54508 y Fs(m;d)10701 54309 y Fn(P)11539 54508 y Fs(u)12139 54309 y Fn( )48 b Fm(k)369 b Fw(=)f Fm(k)p Fn( )48 b Fm(k)434 b Fw(and,)f(since)h Fm(k)p Fn(K)25950 54508 y Fs(m;d)27582 54309 y Fn(P)28420 54508 y Fs(u)29021 54309 y Fm(k)369 b Fw(=)g(1,)434 b(one)f(concludes)h(that)20980 57242 y Fn(K)22087 57441 y Fs(m;d)23719 57242 y Fn(P)24557 57441 y Fs(u)25158 57242 y Fn( )416 b Fw(=)369 b Fn( )48 b(:)13952 b Fw(\(B.28\))3432 60176 y(Therefore)470 b Fn( )517 b Fw(is)470 b(of)h(the)e(form)i(\(\014nite)432 b(p)36 b(olynomial)r(\))320 b Fm(\002)g Fn(e)31864 59694 y Fi(\000)p Fs(x=)p Fq(2)34121 60176 y Fw(,)480 b(whic)-36 b(h)469 b(cannot)h(v)-72 b(anish)3432 61781 y(iden)-36 b(tically)434 b(on)g([0)p Fn(;)221 b(u)p Fw(\).)579 b(Th)-36 b(us)433 b(the)g(con)-36 b(tradiction.)5383 63386 y(T)-108 b(o)576 b(establish)h(the)f(second)g(claim)i(w)-36 b(e)576 b(use)h(that)f Fn( )31536 63585 y Fs(a)32667 63386 y Fw(is)h(eigenfunction)f(of)h Fn(R)588 b Fw(to)3432 64991 y(obtain)4096 68238 y(\()p Fn(P)5440 68437 y Fs(u)6040 68238 y Fw(\()p Fl(1)96 b Fm(\000)g Fn(K)9651 68437 y Fs(m;d)11287 68238 y Fw(\))p Fn( )12639 68437 y Fs(a)13194 68238 y Fw(\)\()p Fn(x)p Fw(\))368 b(=)h(\002\()p Fn(x)96 b Fm(\000)g Fn(u)p Fw(\))21928 66764 y Fj(\020)22721 68238 y Fn(e)23324 67690 y Fi(\000)p Fs(ax)25236 68238 y Fm(\000)g Fn(Z)f Fw(\()p Fm(\000)p Fn(a)p Fw(\))30299 66430 y Fj(Z)31038 69437 y Fk(R)31825 69556 y Ff(\000)32791 68238 y Fw(d)p Fn(y)48 b(L)p Fw(\()p 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b(y)434 b Fn(\014)443 b Fm(2)369 b Fw(\(1)p Fn(=)p Fw(2)296 b Fm(\000)f Fn(\032)g Fw(+)g Fn(a;)221 b Fw(1)296 b Fm(\000)f Fn(\032)p Fw(\).)578 b(Then,)434 b(for)g Fn(u)369 b(>)g Fw(0,)12784 12371 y Fj(\014)12784 13168 y(\014)13227 13500 y Fw(\()p Fn(P)14571 13699 y Fs(u)15171 13500 y Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)19184 13699 y Fs(m;d)20817 13500 y Fw(\))p Fn( )22169 13699 y Fs(a)22723 13500 y Fw(\)\()p Fn(x)p Fw(\))24980 12371 y Fj(\014)24980 13168 y(\014)25792 13500 y Fm(\024)369 b Fw(\002\()p Fn(x)p Fw(\))p Fn(e)30560 12952 y Fi(\000)p Fs(ax)32671 13500 y Fw(+)9325 b(\(B.30\))12784 16216 y(+\002\()p Fn(x)p Fw(\))p Fn(Z)95 b Fw(\()p Fm(\000)p Fn(a)p Fw(\))p Fn(C)21203 16415 y Fs(m;d)22835 16216 y Fn(e)23438 15668 y Fi(\000)p Fq(\()p Fs(\014)46 b Fi(\000)25975 15306 y Fg(1)p 25974 15467 406 40 v 25974 16015 a(2)26513 15668 y Fq(+)p Fs(\032)p Fq(\))p Fs(x)28899 14408 y Fj(Z)29637 17415 y Fk(R)30424 17534 y Ff(\000)31390 16216 y Fw(d)p Fn(y)i(e)33398 15668 y Fs(y)32 b Fq(\()p Fs(\014)46 b Fq(+)p Fs(\032)p Fi(\000)36914 15306 y Fg(1)p 36911 15467 V 36911 16015 a(2)37450 15668 y Fi(\000)p Fs(a)p Fq(\))39103 16216 y Fn(:)3432 19878 y(\014)328 b Fm(\000)5915 19355 y Fq(1)p 5915 19573 471 54 v 5915 20336 a(2)6773 19878 y Fw(+)255 b Fn(\032)369 b(>)f Fw(0,)419 b(b)36 b(ecause)413 b Fn(a)369 b(>)g Fw(0,)418 b(and)c Fn(\014)328 b Fw(+)255 b Fn(\032)g Fm(\000)27729 19355 y Fq(1)p 27729 19573 V 27729 20336 a(2)28588 19878 y Fm(\000)g Fn(a)368 b(>)h Fw(0,)418 b(b)36 b(ecause)414 b Fn(\014)443 b(>)41221 19355 y Fq(1)p 41221 19573 V 41221 20336 a(2)42079 19878 y Fm(\000)255 b Fn(\032)f Fw(+)h Fn(a)p Fw(.)3432 21483 y(Th)-36 b(us)588 b Fn(P)7720 21682 y Fs(u)8320 21483 y Fw(\()p Fl(1)404 b Fm(\000)d Fn(K)12544 21682 y Fs(m;d)14177 21483 y Fw(\))p Fn( )15529 21682 y Fs(a)16717 21483 y Fm(2)632 b Fn(L)19120 21001 y Fq(2)19646 21483 y Fw(\()p Fl(R)p Fw(\))592 b(with)c(norm)h(uniformly)g(b)36 b(ounded)587 b(in)i Fn(u)p Fw(.)1044 b(The)3432 23088 y(second)433 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Fn(\031)11682 29836 y Fw(~)11386 30172 y Fn(C)12317 30371 y Fs(m;d)13949 30172 y Fn(e)14552 29690 y Fi(\000)p Fs(\014)46 b(z)16859 30172 y Fw(for)471 b(an)-36 b(y)471 b(0)433 b Fn(<)g(\014)506 b(<)432 b(\032)p Fw(.)691 b(Th)-36 b(us,)480 b(b)-36 b(y)471 b(c)-36 b(ho)36 b(osing)471 b Fn(\014)507 b Fm(2)432 b Fw(\()p Fn(\032)320 b Fw(+)g Fn(a)h Fm(\000)3432 31777 y Fw(1)p Fn(=)p Fw(2)p Fn(;)221 b(\032)p Fw(\),)435 b(for)f Fn(u)369 b(>)g Fw(0,)12784 33300 y Fj(\014)12784 34097 y(\014)13227 34429 y Fw(\()p Fn(P)14571 34628 y Fs(u)15171 34429 y Fw(\()p Fl(1)299 b Fm(\000)c Fn(K)19184 34628 y Fs(m;d)20817 34429 y Fw(\))p Fn( )22169 34628 y Fs(a)22723 34429 y Fw(\)\()p Fn(x)p Fw(\))24980 33300 y Fj(\014)24980 34097 y(\014)25792 34429 y Fm(\024)369 b Fw(\002\()p Fn(x)p Fw(\))p Fn(e)30560 33881 y Fi(\000)p Fs(ax)32671 34429 y Fw(+)9325 b(\(B.32\))12784 37145 y(+\002\()p Fn(x)p Fw(\))p Fn(Z)95 b Fw(\()p Fm(\000)p Fn(a)p Fw(\))20568 36809 y(~)20272 37145 y Fn(C)21203 37344 y Fs(m;d)22835 37145 y Fn(e)23438 36597 y Fi(\000)p Fq(\()p Fs(\014)46 b Fq(+)25975 36234 y Fg(1)p 25974 36395 406 40 v 25974 36944 a(2)26513 36597 y Fi(\000)p Fs(\032)p Fq(\))p Fs(x)28899 35337 y Fj(Z)29637 38344 y Fk(R)30424 38463 y Ff(\000)31390 37145 y Fw(d)p Fn(y)i(e)33398 36597 y Fs(y)32 b Fq(\()p Fs(\014)46 b Fi(\000)p Fs(\032)p Fq(+)36914 36234 y Fg(1)p 36911 36395 V 36911 36944 a(2)37450 36597 y Fi(\000)p Fs(a)p Fq(\))39103 37145 y Fn(;)3432 40766 y Fw(whic)-36 b(h,)422 b(b)-36 b(y)418 b(the)h(c)-36 b(hoice)419 b(of)h Fn(\014)493 b Fw(and)418 b(since)h Fn(a)369 b(>)f Fw(0,)423 b(implies)c Fn(P)32855 40965 y Fs(u)33456 40766 y Fw(\()p Fl(1)268 b Fm(\000)e Fn(K)37409 40965 y Fs(m;d)39041 40766 y Fw(\))39547 40284 y Fi(\003)40073 40766 y Fn( )40919 40965 y Fs(a)41843 40766 y Fm(2)368 b Fn(L)43982 40284 y Fq(2)44508 40766 y Fw(\()p Fl(R)p Fw(\))3432 42371 y(with)433 b(norm)g(uniformly)i(b)36 b(ounded)432 b(in)h Fn(u)p Fw(.)p 45663 44641 45 878 v 45708 43807 781 45 v 45708 44641 V 46487 44641 45 878 v 3432 49036 a Fp(C)2152 b(The)716 b(Laguerre)g(k)-60 b(ernel)3432 52177 y Fw(Let)399 b Fn(L)6613 51498 y Fq(\()p Fs(\013)p Fq(\))6613 52314 y Fs(n)8404 52177 y Fw(b)36 b(e)399 b(the)h(standard)e Fn(n)p Fw(-th)h(Laguerre)h(p)36 b(olynomial)401 b(of)f(in)-36 b(teger)400 b(order)f Fn(\013)8 b Fw(,)408 b Fn(\013)378 b Fm(\025)369 b Fw(0)3432 53782 y([15)q(],)472 b(Chapter)464 b(5.3.)672 b(The)464 b(Laguerre)g(p)36 b(olynomials)466 b(are)f(orthogonal)g(on)f Fl(R)41157 53981 y Fq(+)42412 53782 y Fw(relativ)-36 b(e)3432 55387 y(to)433 b(the)g(w)-36 b(eigh)g(t)434 b Fn(x)12119 54905 y Fs(\013)12779 55387 y Fn(e)13382 54905 y Fi(\000)p Fs(x)15132 55387 y Fw(as)4780 57235 y Fj(Z)5518 60242 y Fk(R)6305 60361 y Fg(+)7253 59043 y Fw(d)p Fn(xx)9454 58495 y Fs(\013)10113 59043 y Fn(e)10716 58495 y Fi(\000)p Fs(x)12255 57170 y Fj(\022)15090 58145 y Fn(n)p Fw(!)p 13365 58738 4588 54 v 13365 59954 a(\()p Fn(n)296 b Fw(+)e Fn(\013)8 b Fw(\)!)18085 57170 y Fj(\023)19063 57468 y Fq(1)p Fs(=)p Fq(2)20751 59043 y Fn(L)21636 58495 y Fq(\()p Fs(\013)p Fq(\))21636 59372 y Fs(n)23028 59043 y Fw(\()p Fn(x)p Fw(\))25000 57170 y Fj(\022)27835 58145 y Fn(m)p Fw(!)p 26110 58738 4949 54 v 26110 59954 a(\()p Fn(m)295 b Fw(+)g Fn(\013)8 b Fw(\)!)31191 57170 y Fj(\023)32169 57468 y Fq(1)p Fs(=)p Fq(2)33857 59043 y Fn(L)34742 58495 y Fq(\()p Fs(\013)p Fq(\))34742 59372 y Fs(m)36134 59043 y Fw(\()p Fn(x)p Fw(\))369 b(=)f Fn(\016)40210 59242 y Fs(n;m)41930 59043 y Fn(:)1349 b Fw(\(C.1\))3432 62534 y(The)519 b(Laguerre)f(k)-36 b(ernel)520 b(is)f(the)g(orthogonal)g(pro) 72 b(jection)520 b(on)-36 b(to)519 b(the)g(\014rst)f Fn(n)h Fw(Laguerre)3432 64139 y(p)36 b(olynomials)435 b(and)e(is)h(giv)-36 b(en)434 b(b)-36 b(y)7288 67919 y Fn(K)8490 67371 y Fq(\()p Fs(\013)p Fq(\))8395 68247 y Fs(n)9882 67919 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))369 b(=)14719 66259 y Fs(n)p Fi(\000)p Fq(1)14647 66657 y Fj(X)14789 69457 y Fs(j)51 b Fq(=0)18644 67020 y Fn(j)75 b Fw(!)p 16920 67614 4424 54 v 16920 68830 a(\()p Fn(j)369 b Fw(+)295 b Fn(\013)8 b Fw(\)!)21476 67919 y Fn(L)22361 67240 y Fq(\()p Fs(\013)p Fq(\))22361 68260 y Fs(j)23752 67919 y Fw(\()p Fn(x)p Fw(\))p Fn(L)26388 67240 y Fq(\()p Fs(\013)p Fq(\))26388 68260 y Fs(j)27780 67919 y Fw(\()p Fn(y)48 b Fw(\))p Fn(x)30213 67371 y Fs(\013=)p Fq(2)31813 67919 y Fn(y)32495 67371 y Fs(\013=)p Fq(2)34095 67919 y Fn(e)34698 67371 y Fi(\000)p Fs(x=)p Fq(2)36956 67919 y Fn(e)37559 67371 y Fi(\000)p Fs(y)32 b(=)p Fq(2)43640 67919 y Fw(\(C.2\))3432 71842 y(for)434 b Fn(x;)221 b(y)417 b Fm(\025)369 b Fw(0.)24367 75163 y(46)p eop end end %%Page: 47 47 TeXDict begin HPSdict begin 47 46 bop 3432 7638 a Fh(Prop)42 b(osition)499 b(C.1.)554 b Fo(With)464 b(de\014nitions)f(\(3.13\))j (and)f(\(C.2\))g(one)g(has)18022 10405 y Fn(K)19129 10604 y Fs(m;)p Fi(\000)p Fs(d)21493 10405 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))369 b(=)g Fn(K)27365 10604 y Fs(m;d)28998 10405 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))11627 b(\(C.3\))3432 13172 y Fo(and)16587 15075 y Fn(K)17694 15274 y Fs(m;d)19327 15075 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))369 b(=)f Fn(K)25293 14396 y Fq(\(2)p Fs(d)p Fq(\))25198 15445 y Fs(m)p Fi(\000)p Fs(d)27302 15075 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))30317 13600 y Fj(\020)31243 14176 y Fn(x)p 31243 14769 740 54 v 31272 15986 a(y)32115 13600 y Fj(\021)32908 13898 y Fs(d)43640 15075 y Fw(\(C.4\))3432 17926 y Fo(for)464 b Fw(0)369 b Fm(\024)h Fn(d)e(<)h(m)465 b Fo(and)g Fn(x;)221 b(y)417 b(>)369 b Fw(0)p Fo(.)3432 20495 y(Pr)-66 b(o)g(of.)649 b Fw(Let)433 b(us)g(de\014ne)14228 23469 y(^)-696 b Fn(g)14805 23668 y Fs(`)15245 23469 y Fw(\()p Fn(k)45 b Fw(\))1106 b(=)20203 22393 y Fj(\000)20944 22946 y Fq(1)p 20944 23163 471 54 v 20944 23927 a(2)21843 23469 y Fm(\000)295 b Fn(ik)24336 22393 y Fj(\001)24945 22690 y Fs(m)p Fi(\000)p Fs(d)27048 22393 y Fj(\000)27790 22946 y Fq(1)p 27790 23163 V 27790 23927 a(2)28689 23469 y Fw(+)f Fn(ik)31160 22393 y Fj(\001)31769 22690 y Fi(\000)p Fq(\()p Fs(m)p Fq(+)p Fs(d)p Fq(\))35336 23469 y Fn(;)14162 25819 y Fw(^)-696 b Fn(g)14739 26018 y Fs(r)15245 25819 y Fw(\()p Fn(k)45 b Fw(\))1106 b(=)20203 24743 y Fj(\000)20944 25296 y Fq(1)p 20944 25514 V 20944 26277 a(2)21843 25819 y Fm(\000)295 b Fn(ik)24336 24743 y Fj(\001)24945 25041 y Fs(m)p Fq(+)p Fs(d)27048 24743 y Fj(\000)27790 25296 y Fq(1)p 27790 25514 V 27790 26277 a(2)28689 25819 y Fw(+)f Fn(ik)31160 24743 y Fj(\001)31769 25041 y Fi(\000)p Fq(\()p Fs(m)p Fi(\000)p Fs(d)p Fq(\))35336 25819 y Fn(:)7943 b Fw(\(C.5\))3432 28586 y(W)-108 b(e)433 b(set)h Fn(\013)377 b Fw(=)369 b(2)p Fn(d)p Fw(,)434 b Fn(n)369 b Fw(=)g Fn(m)295 b Fm(\000)g Fn(d)g Fm(\000)h Fw(1.)578 b(By)434 b([15)q(],)h(p.)e(244,)i(it)e(follo)-36 b(ws)4461 30204 y Fj(Z)5199 33211 y Fk(R)6257 32012 y Fw(d)p Fn(k)45 b(e)8304 31464 y Fs(ik)24 b(x)9769 32012 y Fw(^)-696 b Fn(g)10346 32211 y Fs(`)10785 32012 y Fw(\()p Fn(k)45 b Fw(\))369 b(=)g Fn(g)14891 32211 y Fs(`)15330 32012 y Fw(\()p Fn(x)p Fw(\))g(=)f(\()p Fm(\000)p Fw(1\))21525 31464 y Fs(n)24009 31113 y Fn(n)p Fw(!)p 22285 31707 4588 54 v 22285 32923 a(\()p Fn(n)295 b Fw(+)g Fn(\013)8 b Fw(\)!)27005 32012 y Fn(e)27608 31464 y Fi(\000)p Fs(x=)p Fq(2)29865 32012 y Fw(\(1)295 b Fm(\000)h Fn(@)33332 32211 y Fs(x)33916 32012 y Fw(\))p Fn(x)35161 31464 y Fs(\013)35821 32012 y Fn(L)36706 31464 y Fq(\()p Fs(\013)p Fq(\))36706 32340 y Fs(n)38097 32012 y Fw(\()p Fn(x)p Fw(\)\002\()p Fn(x)p Fw(\))1029 b(\(C.6\))3432 35450 y(and)17578 35561 y Fj(Z)18316 38568 y Fk(R)19374 37369 y Fw(d)p Fn(k)45 b(e)21421 36821 y Fi(\000)p Fs(ik)24 b(x)23617 37369 y Fw(^)-696 b Fn(g)24194 37568 y Fs(r)24700 37369 y Fw(\()p Fn(k)45 b Fw(\))369 b(=)f Fn(g)28805 37568 y Fs(r)29311 37369 y Fw(\()p Fm(\000)p Fn(x)p Fw(\))p Fn(;)11184 b Fw(\(C.7\))3432 40250 y(with)7495 43017 y Fn(g)8118 43216 y Fs(r)8624 43017 y Fw(\()p Fn(x)p Fw(\))369 b(=)12124 41941 y Fj(\000)12866 42494 y Fq(1)p 12866 42711 471 54 v 12866 43475 a(2)13764 43017 y Fw(+)295 b Fn(@)15758 43216 y Fs(x)16343 41941 y Fj(\001)16952 42238 y Fs(d)17491 41941 y Fj(\000)18233 42494 y Fq(1)p 18233 42711 V 18233 43475 a(2)19131 43017 y Fm(\000)h Fn(@)21147 43216 y Fs(x)21731 41941 y Fj(\001)22340 42238 y Fs(d)p Fq(+1)24082 43017 y Fw(\()p Fm(\000)p Fw(1\))26777 42468 y Fs(m)p Fq(+1)28867 43017 y Fn(e)29470 42468 y Fi(\000)p Fs(x=)p Fq(2)31728 43017 y Fn(L)32613 42338 y Fq(\(0\))32613 43343 y Fs(m)p Fi(\000)p Fq(1)34702 43017 y Fw(\()p Fn(x)p Fw(\)\002\()p Fn(x)p Fw(\))p Fn(:)4063 b Fw(\(C.8\))3432 45784 y Fn(g)4055 45983 y Fs(`)4920 45784 y Fw(is)427 b(supp)36 b(orted)425 b(on)h([0)p Fn(;)221 b Fm(1)p Fw(\))427 b(with)g(a)f(discon)-36 b(tin)g(uit)g(y)426 b(at)g Fn(x)369 b Fw(=)g(0.)576 b(As)426 b(a)h(distribution)e Fn(g)46097 45983 y Fs(r)3432 47389 y Fw(is)433 b(supp)36 b(orted)433 b(in)g([0)p Fn(;)221 b Fm(1)p Fw(\))435 b(with)f(the)f (singular)h(part)f(concen)-36 b(trated)432 b(at)i Fm(f)p Fn(x)369 b Fw(=)g(0)p Fm(g)p Fw(.)5383 48994 y(By)434 b([15)q(],)g(Section)f(5.5.2,)j(one)d(has)h(the)f(iden)-36 b(tities)12189 51942 y(\(1)296 b Fm(\000)f Fn(@)15656 52141 y Fs(x)16241 51942 y Fw(\))p Fn(L)17632 51393 y Fq(\()p Fs(\013)p Fq(\))17632 52270 y Fs(n)19392 51942 y Fw(=)369 b Fm(\000)p Fn(@)22493 52141 y Fs(x)23077 51942 y Fn(L)23962 51263 y Fq(\()p Fs(\013)p Fq(\))23962 52268 y Fs(n)p Fq(+1)25791 51942 y Fn(;)1522 b(@)28361 52141 y Fs(x)28946 51942 y Fn(L)29831 51393 y Fq(\()p Fs(\013)p Fq(\))29831 52270 y Fs(n)31591 51942 y Fw(=)369 b Fm(\000)p Fn(L)34890 51263 y Fq(\()p Fs(\013)p Fq(+1\))34890 52268 y Fs(n)p Fq(+1)37484 51942 y Fn(:)5795 b Fw(\(C.9\))3432 54709 y(Using)433 b(them)g(rep)36 b(eatedly)434 b(in)g(\(C.6\))g(and)f (\(C.8\))h(yields,)h(for)f Fn(x)369 b(>)f Fw(0,)6796 58135 y Fn(g)7419 58334 y Fs(`)7859 58135 y Fw(\()p Fn(x)p Fw(\))h(=)f(\()p Fm(\000)p Fw(1\))14054 57586 y Fs(n)p Fq(+1)17740 57236 y Fn(n)p Fw(!)p 16016 57829 4588 54 v 16016 59046 a(\()p Fn(n)295 b Fw(+)g Fn(\013)8 b Fw(\)!)20736 58135 y Fn(e)21339 57586 y Fi(\000)p Fs(x=)p Fq(2)23596 57059 y Fj(\000)24205 58135 y Fn(x)24944 57586 y Fs(\013)25604 58135 y Fn(@)26291 58334 y Fs(x)26876 58135 y Fn(L)27761 57456 y Fq(\()p Fs(\013)p Fq(\))27761 58460 y Fs(n)p Fq(+1)29589 58135 y Fw(\()p Fn(x)p Fw(\))295 b(+)g Fn(\013)8 b(x)34516 57586 y Fs(\013)p Fi(\000)p Fq(1)36379 58135 y Fn(L)37264 57586 y Fq(\()p Fs(\013)p Fq(\))37264 58463 y Fs(n)38655 57059 y Fj(\001)39264 58135 y Fn(;)3365 b Fw(\(C.10\))3432 61573 y(and)15962 63178 y Fn(g)16585 63377 y Fs(r)17091 63178 y Fw(\()p Fn(x)p Fw(\))369 b(=)f(\()p Fm(\000)p Fw(1\))23286 62630 y Fs(n)p Fq(+1)25115 63178 y Fn(e)25718 62630 y Fi(\000)p Fs(x=)p Fq(2)27975 63178 y Fn(@)28662 63377 y Fs(x)29247 63178 y Fn(L)30132 62499 y Fq(\()p Fs(\013)p Fq(\))30132 63504 y Fs(n)p Fq(+1)31961 63178 y Fw(\()p Fn(x)p Fw(\))p Fn(:)8917 b Fw(\(C.11\))5383 65432 y(With)433 b(these)g(notations)h(the)f(in)-36 b(tegral)434 b(k)-36 b(ernel)434 b(from)g(\(3.13\))g(is)g(expressed)f(as)8394 67303 y Fj(\000)9003 68379 y Fn(T)9946 67831 y Fs(m)p Fq(+)p Fs(d)9765 68708 y Fq(+)12050 68379 y Fn(T)12993 67700 y Fi(\000)p Fq(\()p Fs(m)p Fi(\000)p Fs(d)p Fq(\))12812 68669 y Fi(\000)16560 68379 y Fn(P)17398 68578 y Fi(\000)18185 68379 y Fn(T)19128 67700 y Fi(\000)p Fq(\()p Fs(m)p Fq(+)p Fs(d)p Fq(\))18947 68669 y(+)22696 68379 y Fn(T)23639 67831 y Fs(m)p Fi(\000)p Fs(d)23458 68708 y Fi(\000)25742 67303 y Fj(\001)26351 68379 y Fw(\()p Fn(x;)221 b(y)48 b Fw(\))13624 b(\(C.12\))8394 71095 y(=)9775 69287 y Fj(Z)10513 72294 y Fk(R)11300 72413 y Ff(\000)12266 71095 y Fw(d)p Fn(w)36 b(g)14578 71294 y Fs(`)15016 71095 y Fw(\()p Fn(x)295 b Fm(\000)h Fn(w)36 b Fw(\))p Fn(g)19980 71294 y Fs(r)20485 71095 y Fw(\()p Fm(\000)p Fw(\()p Fn(w)330 b Fm(\000)296 b Fn(y)48 b Fw(\)\))368 b(=)28562 69287 y Fj(Z)29300 72294 y Fk(R)30087 72413 y Fg(+)31034 71095 y Fw(d)p Fn(w)36 b(g)33346 71294 y Fs(`)33785 71095 y Fw(\()p Fn(x)295 b Fw(+)g Fn(w)36 b Fw(\))p Fn(g)38727 71294 y Fs(r)39232 71095 y Fw(\()p Fn(y)343 b Fw(+)295 b Fn(w)36 b Fw(\))p Fn(:)24367 75163 y Fw(47)p eop end end %%Page: 48 48 TeXDict begin HPSdict begin 48 47 bop 3432 7638 a Fw(Note)473 b(that)f(for)i Fn(x;)221 b(y)485 b(>)436 b Fw(0)473 b(only)h(the)e (regular)i(part)e(of)i Fn(g)31131 7837 y Fs(r)32110 7638 y Fw(is)f(used.)696 b(W)-108 b(e)473 b(insert)g(\(C.10\))3432 9243 y(and)433 b(\(C.11\))h(in)g(\(C.12\).)579 b(Then)433 b(\(C.4\))h(of)h(Prop)36 b(osition)434 b(C.1)g(amoun)-36 b(ts)433 b(to)7304 10799 y Fs(n)6630 11197 y Fj(X)6772 13997 y Fs(j)51 b Fq(=0)10627 11561 y Fn(j)75 b Fw(!)p 8903 12154 4424 54 v 8903 13371 a(\()p Fn(j)370 b Fw(+)294 b Fn(\013)8 b Fw(\)!)13459 12459 y Fn(x)14198 11911 y Fs(\013)14858 12459 y Fn(L)15743 11781 y Fq(\()p Fs(\013)p Fq(\))15743 12800 y Fs(j)17134 12459 y Fw(\()p Fn(x)p Fw(\))p Fn(L)19770 11781 y Fq(\()p Fs(\013)p Fq(\))19770 12800 y Fs(j)21161 12459 y Fw(\()p Fn(y)48 b Fw(\))369 b(=)26462 11561 y Fn(n)p Fw(!)p 24737 12154 4588 54 v 24737 13371 a(\()p Fn(n)296 b Fw(+)f Fn(\013)8 b Fw(\)!)29679 10651 y Fj(Z)30417 13659 y Fk(R)31204 13778 y Fg(+)32151 12459 y Fw(d)p Fn(w)36 b(e)34443 11911 y Fi(\000)p 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Fw(\))p Fn(L)35133 36196 y Fi(0)35133 37073 y Fs(n)35759 36744 y Fw(\()p Fn(y)342 b Fw(+)295 b Fn(w)36 b Fw(\))40020 35270 y Fj(\021)40813 36744 y Fn(;)1816 b Fw(\(C.15\))3432 39288 y(omitting)650 b(the)f(sup)36 b(erscript)649 b Fn(\013)8 b Fw(.)1227 b(In)650 b(the)f(second)g(in)-36 b(tegral)651 b(w)-36 b(e)650 b(use)f(the)g(iden)-36 b(tities)3432 40893 y Fn(L)4317 40411 y Fi(0)4317 41221 y Fs(n)p Fq(+1)6514 40893 y Fw(=)369 b Fn(L)8780 40411 y Fi(0)8780 41221 y Fs(n)9679 40893 y Fm(\000)273 b Fn(L)11870 41092 y Fs(n)12918 40893 y Fw(and)422 b Fn(xL)17060 40411 y Fi(0)17060 41221 y Fs(n)18056 40893 y Fw(=)368 b Fn(nL)21097 41092 y Fs(n)21997 40893 y Fm(\000)273 b Fw(\()p Fn(n)g Fw(+)f Fn(\013)8 b Fw(\))p Fn(L)28368 41092 y Fs(n)p Fi(\000)p Fq(1)30197 40893 y Fw(.)575 b(Then)422 b(the)g(terms)g(com)-36 b(bine)423 b(as)5023 43831 y Fn(A)5998 44030 y Fs(n)6919 43831 y Fm(\000)8380 42933 y Fn(n)296 b Fw(+)f Fn(\013)p 8380 43526 V 9599 44742 a(n)11727 43831 y(A)12702 44030 y Fs(n)p 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58258 45 878 v 45708 57424 781 45 v 45708 58258 V 46487 58258 45 878 v 3432 62588 a Fp(D)2152 b(Some)717 b(useful)e(relations)3432 65895 y Fd(D.1)1793 b(Tw)-50 b(o)598 b(represen)-50 b(tations)601 b(of)d(the)g(Airy)h(functions)3432 68358 y Fw(F)-108 b(or)433 b(an)-36 b(y)434 b Fn(\033)416 b(<)369 b Fw(0,)434 b(de\014ne)f(the)g(path)g Fn(\015)21908 68557 y Fs(\033)22901 68358 y Fw(=)369 b Fn(\033)343 b Fw(+)295 b Fn(i)p Fl(R)p Fw(.)581 b(Then)17436 71134 y(Ai\()p Fn(z)59 b Fw(\))369 b(=)22944 70236 y(1)p 22330 70829 1880 54 v 22330 72046 a(2)p Fn(\031)48 b(i)24563 69326 y Fj(Z)25301 72334 y Fs(\015)25786 72445 y Fe(\033)26618 71134 y Fw(d)p Fn(\030)64 b(e)28576 70586 y Fi(\000)29441 70224 y Fg(1)p 29440 70385 406 40 v 29440 70933 a(3)29979 70586 y Fs(\030)30428 70273 y Fg(3)30888 70586 y Fq(+)p Fs(z)37 b(\030)43586 71134 y Fw(\(D.1\))24367 75163 y(48)p eop end end %%Page: 49 49 TeXDict begin HPSdict begin 49 48 bop 3432 7638 a Fw(and)433 b(for)h(an)-36 b(y)434 b Fn(\026)368 b(>)h 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52438 406 40 v 27002 52987 a(3)27540 52640 y Fs(\030)27989 52327 y Fg(3)28450 52640 y Fq(+)29315 52277 y Fg(1)p 29314 52438 V 29314 52987 a(3)29853 52640 y Fs(w)30552 52327 y Fg(3)31013 52640 y Fq(+\()p Fs(w)32810 52327 y Fg(2)33271 52640 y Fq(+)p Fs(s)p Fq(\)\()p Fs(\030)35 b Fi(\000)p Fs(w)24 b Fq(\))p 25534 53723 11937 54 v 29123 54940 a Fw(\()p Fn(\030)359 b Fm(\000)296 b Fn(w)36 b Fw(\))33357 54556 y Fq(2)37603 54028 y Fw(;)221 b Fn(\030)434 b Fw(=)369 b Fn(w)41534 52155 y Fj(\023)42880 54028 y Fw(=)g Fn(s:)43586 56445 y Fw(\(D.6\))3432 60275 y Fd(D.3)1793 b(Tw)-50 b(o)598 b(equiv)-100 b(alen)-50 b(t)601 b(expressions)f(for)e Fc(S)33674 60514 y Fb(w)30 b(;s)3432 62738 y Fw(Let)429 b(us)h(consider)g Fn(w)404 b(>)369 b Fw(0.)577 b(The)430 b(represen)-36 b(tation)429 b(of)i(the)f(Airy)g(function)g(\(D.2\))g (allo)-36 b(ws)3432 64343 y(one)433 b(to)h(write)6457 67475 y Fn(S)7257 67674 y Fs(w)24 b(;s)9815 67475 y Fw(=)1106 b Fn(s)295 b Fw(+)g Fn(e)14751 66927 y Fi(\000)15616 66565 y 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b(e)32281 70046 y Fg(1)p 32281 70207 406 40 v 32281 70755 a(3)32819 70408 y Fs(\030)33268 70095 y Fg(3)33784 70956 y Fn(e)34387 70408 y Fq(\()p Fs(w)35452 70095 y Fg(2)35913 70408 y Fq(+)p Fs(s)p Fq(+)p Fs(x)p Fq(+)p Fs(y)32 b Fq(\)\()p Fs(w)24 b Fi(\000)p Fs(\030)35 b Fq(\))43586 70956 y Fw(\(D.7\))24367 75163 y(49)p eop end end %%Page: 50 50 TeXDict begin HPSdict begin 50 49 bop 3432 7638 a Fw(By)621 b(c)-36 b(ho)36 b(osing)622 b Fn(\026)687 b Fw(=)g(2)p Fn(w)36 b Fw(,)668 b(one)621 b(obtains)g(an)g(in)-36 b(tegrand)620 b(whic)-36 b(h)621 b(is)g(absolutely)h(in)-36 b(te-)3432 9243 y(grable.)726 b(Th)-36 b(us)482 b(b)-36 b(y)482 b(F)-108 b(ubini's)482 b(Theorem)h(w)-36 b(e)483 b(are)f(allo)-36 b(w)g(ed)484 b(to)f(exc)-36 b(hange)483 b(the)f(order)g(of)3432 10848 y(the)433 b(in)-36 b(tegrals)434 b(and)f(compute)g(\014rst)f(the)h(one)h(in)f(\()p Fn(x;)221 b(y)48 b Fw(\))434 b(with)g(the)f(result)14031 14546 y Fn(S)14831 14745 y Fs(w)24 b(;s)16651 14546 y Fw(=)368 b Fn(s)295 b Fw(+)20246 12738 y Fj(Z)20984 15745 y Fs(\015)21521 15440 y Fe(r)21469 16027 y Fg(2)p Fe(w)22817 14546 y Fw(d)p Fn(\030)24305 13647 y(e)25041 12795 y Fg(1)p 25041 12956 406 40 v 25041 13504 a(3)25580 13157 y Fs(\030)26029 12844 y Fg(3)26489 13157 y Fi(\000)27354 12795 y Fg(1)p 27354 12956 V 27354 13504 a(3)27892 13157 y Fs(w)28591 12844 y Fg(3)29053 13157 y Fq(+\()p Fs(w)30850 12844 y Fg(2)31311 13157 y Fq(+)p Fs(s)p Fq(\)\()p Fs(w)24 b Fi(\000)p Fs(\030)35 b Fq(\))p 24305 14240 11205 54 v 26588 15457 a Fw(2)p Fn(\031)48 b(i)p Fw(\()p Fn(w)330 b Fm(\000)296 b Fn(\030)64 b Fw(\))32701 15073 y Fq(2)35643 14546 y Fn(:)7582 b Fw(\(D.8\))3432 18210 y(Let)433 b(\000)6575 18409 y Fs(w)7762 18210 y Fw(b)36 b(e)434 b(as)g(in)f(\(D.5\),)h(then) 4913 21907 y Fn(S)5713 22106 y Fs(w)24 b(;s)8271 21907 y Fw(=)1107 b Fn(s)295 b Fw(+)12604 20099 y Fj(Z)13342 23107 y Fs(\015)13879 22801 y Fe(r)13827 23459 y(w)18 b(=)p Fg(2)15600 21907 y Fw(d)p Fn(\030)17088 21009 y(e)17824 20156 y Fg(1)p 17824 20317 406 40 v 17824 20866 a(3)18362 20518 y Fs(\030)18811 20206 y Fg(3)19272 20518 y Fi(\000)20137 20156 y Fg(1)p 20137 20317 V 20137 20866 a(3)20675 20518 y Fs(w)21374 20206 y Fg(3)21835 20518 y Fq(+\()p Fs(w)23632 20206 y Fg(2)24093 20518 y Fq(+)p Fs(s)p Fq(\)\()p Fs(w)24 b Fi(\000)p Fs(\030)35 b Fq(\))p 17088 21602 11205 54 v 19371 22818 a Fw(2)p Fn(\031)48 b(i)p Fw(\()p Fn(w)330 b Fm(\000)296 b Fn(\030)64 b Fw(\))25484 22435 y Fq(2)28721 21907 y Fw(+)30027 20099 y Fj(Z)30765 23107 y Fq(\000)31353 23218 y Fe(w)32296 21907 y Fw(d)p Fn(\030)33784 21009 y(e)34520 20156 y Fg(1)p 34520 20317 406 40 v 34520 20866 a(3)35058 20518 y Fs(\030)35507 20206 y Fg(3)35968 20518 y Fi(\000)36833 20156 y Fg(1)p 36832 20317 V 36832 20866 a(3)37371 20518 y Fs(w)38070 20206 y Fg(3)38531 20518 y Fq(+\()p Fs(w)40328 20206 y Fg(2)40789 20518 y Fq(+)p Fs(s)p Fq(\)\()p Fs(w)24 b Fi(\000)p Fs(\030)35 b Fq(\))p 33784 21602 11205 54 v 36067 22818 a Fw(2)p Fn(\031)48 b(i)p Fw(\()p Fn(w)330 b Fm(\000)295 b Fn(\030)64 b Fw(\))42179 22435 y Fq(2)8271 26247 y Fw(=)10390 24439 y Fj(Z)11128 27446 y Fs(\015)11665 27141 y Fe(r)11613 27799 y(w)18 b(=)p Fg(2)13386 26247 y Fw(d)p Fn(\030)14874 25348 y(e)15610 24496 y Fg(1)p 15609 24657 406 40 v 15609 25205 a(3)16148 24858 y Fs(\030)16597 24545 y Fg(3)17057 24858 y Fi(\000)17922 24496 y Fg(1)p 17922 24657 V 17922 25205 a(3)18461 24858 y Fs(w)19160 24545 y Fg(3)19621 24858 y Fq(+\()p Fs(w)21418 24545 y Fg(2)21879 24858 y Fq(+)p Fs(s)p Fq(\)\()p Fs(w)24 b Fi(\000)p Fs(\030)35 b Fq(\))p 14873 25941 11205 54 v 17157 27158 a Fw(2)p Fn(\031)48 b(i)p Fw(\()p Fn(w)330 b Fm(\000)295 b Fn(\030)64 b Fw(\))23269 26774 y Fq(2)26211 26247 y Fn(:)17014 b Fw(\(D.9\))3432 30073 y(On)432 b(the)h(other)h (hand,)3920 31518 y Fj(Z)4658 34525 y Fk(R)5445 34220 y Fg(2)5445 34804 y Ff(\000)6411 33326 y Fw(d)p Fn(x)p Fw(d)p Fn(y)48 b Fw(\010)10217 33525 y Fs(w)24 b(;s)11667 33326 y Fw(\()p Fn(x)295 b Fw(+)g Fn(y)48 b Fw(\))1107 b(=)18927 31518 y Fj(Z)19665 34525 y Fk(R)20452 34220 y Fg(2)20452 34804 y Ff(\000)21418 33326 y Fw(d)p Fn(x)p Fw(d)p Fn(y)25032 32427 y Fw(1)p 24417 33021 1880 54 v 24417 34237 a(2)p Fn(\031)48 b(i)26651 31518 y Fj(Z)27389 34525 y Fs(w)24 b(=)p Fq(2+)p Fs(i)p Fk(R)31139 33326 y Fw(d)p Fn(\030)64 b(e)33230 32416 y Fg(1)p 33230 32577 406 40 v 33230 33125 a(3)33768 32778 y Fs(\030)34217 32465 y Fg(3)34678 32778 y Fi(\000)35543 32416 y Fg(1)p 35542 32577 V 35542 33125 a(3)36081 32778 y Fs(w)36780 32465 y Fg(3)37297 33326 y Fn(e)37900 32778 y Fq(\()p Fs(w)24 b Fi(\000)p Fs(\030)35 b Fq(\)\()p Fs(w)41577 32465 y Fg(2)42037 32778 y Fq(+)p Fs(s)p Fq(+)p Fs(x)p Fq(+)p Fs(y)d Fq(\))16809 37518 y Fw(=)18927 35710 y Fj(Z)19665 38717 y Fs(w)24 b(=)p Fq(2+)p Fs(i)p Fk(R)23415 37518 y Fw(d)p Fn(\030)24903 36619 y(e)25639 35767 y Fg(1)p 25639 35928 V 25639 36476 a(3)26177 36129 y Fs(\030)26626 35817 y Fg(3)27087 36129 y Fi(\000)27952 35767 y Fg(1)p 27951 35928 V 27951 36476 a(3)28490 36129 y Fs(w)29189 35817 y Fg(3)29650 36129 y Fq(+\()p Fs(w)31447 35817 y Fg(2)31908 36129 y Fq(+)p Fs(s)p Fq(\)\()p Fs(w)g Fi(\000)p Fs(\030)35 b Fq(\))p 24903 37213 11205 54 v 27186 38429 a Fw(2)p Fn(\031)48 b(i)p Fw(\()p Fn(w)330 b Fm(\000)296 b Fn(\030)64 b Fw(\))33299 38045 y Fq(2)36609 37518 y Fw(=)369 b Fn(S)38790 37717 y Fs(w)24 b(;s)42936 37518 y Fw(\(D.10\))3432 41121 y(b)-36 b(y)433 b(deforming)h(the)f(in)-36 b(tegral)434 b(on)f Fn(w)36 b(=)p Fw(2)296 b(+)f Fn(i)p Fl(R)436 b Fw(to)e Fn(\015)28289 40639 y Fs(r)28217 41516 y(w)24 b(=)p Fq(2)29912 41121 y Fw(.)3432 45521 y Fp(References)4082 48442 y Fw([1])651 b(J.)946 b(Baik,)1075 b(G.)945 b(Ben)g(Arous,)1073 b(and)945 b(S.)g(P)-36 b(\023)-614 b(ec)-36 b(h)g(\023)-614 b(e,)1073 b Fo(Phase)936 b(tr)-66 b(ansition)933 b(of)i(the)6105 50047 y(lar)-66 b(gest)761 b(eigenvalue)g(for)h(non-nul)66 b(l)761 b(c)-66 b(omplex)761 b(sample)h(c)-66 b(ovarianc)g(e)760 b(matric)-66 b(es)p Fw(,)6105 51652 y(arXiv:math.PS/0403022,)437 b(to)d(app)36 b(ear)433 b(in)h(Ann.)f(Probab.)g(\(2004\).)4082 54292 y([2])651 b(J.)406 b(Baik)h(and)d(E.M.)i(Rains,)412 b Fo(Limiting)436 b(distributions)i(for)h(a)g(p)-66 b(olynucle)g(ar)438 b(gr)-66 b(owth)6105 55897 y(mo)g(del)465 b(with)f(external)g(sour)-66 b(c)g(es)p Fw(,)434 b(J.)g(Stat.)f(Ph)-36 b(ys.)434 b Fh(100)g Fw(\(2000\),)h(523{542.)4082 58537 y([3])651 b(J.)593 b(Baik)h(and)f(E.M.)g(Rains,)633 b Fo(Symmetrize)-66 b(d)609 b(r)-66 b(andom)611 b(p)-66 b(ermuations)p Fw(,)632 b(Random)6105 60142 y(Matrix)517 b(Mo)36 b(dels)516 b(and)f(Their)h (Applications,)537 b(v)-36 b(ol.)517 b(40,)537 b(Cam)-36 b(bridge)516 b(Univ)-36 b(ersit)g(y)6105 61747 y(Press,)434 b(2001,)h(pp.)e(1{19.)4082 64387 y([4])651 b(H.)e(v)-72 b(an)648 b(Beijeren,)703 b(R.)649 b(Kutner,)701 b(and)648 b(H.)g(Sp)36 b(ohn,)701 b Fo(Exc)-66 b(ess)663 b(noise)e(for)h(driven) 6105 65992 y(di\013usive)465 b(systems)p Fw(,)434 b(Ph)-36 b(ys.)433 b(Rev.)i(Lett.)e Fh(54)h Fw(\(1985\),)h(2026{2029.)4082 68632 y([5])651 b(F.)525 b(Colaiori)i(and)d(M.A.)h(Mo)36 b(ore,)549 b Fo(Numeric)-66 b(al)547 b(solution)i(of)f(the)h(mo)-66 b(de-c)g(oupling)6105 70237 y(e)g(quations)378 b(for)h(the)f(Kar)-66 b(dar-Parisi-Zhang)378 b(e)-66 b(quation)378 b(in)g(one)g(dimension)p Fw(,)358 b(Ph)-36 b(ys.)6105 71842 y(Rev.)435 b(E)e Fh(65)h Fw(\(2002\),)h(017105.)24367 75163 y(50)p eop end end %%Page: 51 51 TeXDict begin HPSdict begin 51 50 bop 4082 7638 a Fw([6])651 b(P)-108 b(.L.)475 b(F)-108 b(errari,)484 b Fo(Polynucle)-66 b(ar)503 b(gr)-66 b(owth)502 b(on)g(a)h(\015at)f(substr)-66 b(ate)503 b(and)f(e)-66 b(dge)502 b(sc)-66 b(aling)501 b(of)6105 9243 y(GOE)466 b(eigenvalues)p Fw(,)433 b(Comm.)h(Math.)f(Ph) -36 b(ys.)434 b Fh(252)g Fw(\(2004\),)h(77{109.)4082 11955 y([7])651 b(D.)307 b(F)-108 b(orster,)331 b(D.R.)307 b(Nelson,)332 b(and)305 b(M.J.)h(Stephen,)330 b Fo(L)-66 b(ar)g(ge-distanc)g(e)345 b(and)j(long-time)6105 13560 y(pr)-66 b(op)g(erties)424 b(of)h(a)g(r)-66 b(andomly)424 b(stirr)-66 b(e)g(d)424 b(\015uid)p Fw(,)399 b(Ph)-36 b(ys.)390 b(Rev.)i(A)e Fh(16)g Fw(\(1977\),)400 b(732{749.)4082 16272 y([8])651 b(T.)723 b(Imam)-36 b(ura)721 b(and)g(T.)i(Sasamoto,) 794 b Fo(Fluctuations)730 b(of)f(the)h(one-dimensional)6105 17877 y(p)-66 b(olynucle)g(ar)735 b(gr)-66 b(owth)736 b(mo)-66 b(del)736 b(with)g(external)e(sour)-66 b(c)g(es)p Fw(,)803 b(Nucl.)729 b(Ph)-36 b(ys.)728 b(B)g Fh(699)6105 19482 y Fw(\(2004\),)435 b(503{544.)4082 22195 y([9])651 b(K.)494 b(Johansson,)509 b Fo(Shap)-66 b(e)519 b(\015uctuations)h(and) g(r)-66 b(andom)520 b(matric)-66 b(es)p Fw(,)507 b(Comm.)494 b(Math.)6105 23800 y(Ph)-36 b(ys.)434 b Fh(209)g Fw(\(2000\),)h (437{476.)3432 26512 y([10])651 b(K.)399 b(Johansson,)406 b Fo(Discr)-66 b(ete)430 b(p)-66 b(olynucle)g(ar)432 b(gr)-66 b(owth)432 b(and)h(determinantal)d(pr)-66 b(o)g(c)g(esses)p Fw(,)6105 28117 y(Comm.)434 b(Math.)g(Ph)-36 b(ys.)434 b Fh(242)g Fw(\(2003\),)h(277{329.)3432 30829 y([11])651 b(K.)479 b(Kardar,)489 b(G.)478 b(P)-36 b(arisi,)490 b(and)477 b(Y.Z.)i(Zhang,)489 b Fo(Dynamic)504 b(sc)-66 b(aling)505 b(of)g(gr)-66 b(owing)505 b(in-)6105 32434 y(terfac)-66 b(es)p Fw(,)433 b(Ph)-36 b(ys.)433 b(Rev.)i(Lett.)e Fh(56)h Fw(\(1986\),)h(889{892.)3432 35146 y([12])651 b(J.)589 b(Krug,)628 b(P)-108 b(.)588 b(Meakin,)628 b(and)588 b(T.)h(Halpin-Healy)-108 b(,)628 b Fo(A)-33 b(mplitude)606 b(universality)g(for)6105 36751 y(driven)452 b(interfac)-66 b(es)451 b(and)i(dir)-66 b(e)g(cte)g(d)451 b(p)-66 b(olymers)453 b(in)g(r)-66 b(andom)452 b(me)-66 b(dia)p Fw(,)423 b(Ph)-36 b(ys.)421 b(Rev.)h(A)6105 38356 y Fh(45)434 b Fw(\(1992\),)h(638{653.) 3432 41068 y([13])651 b(T.M.)590 b(Liggett,)628 b Fo(Coupling)607 b(the)h(simple)f(exclusion)g(pr)-66 b(o)g(c)g(ess)p Fw(,)628 b(Ann.)588 b(Probab.)h Fh(4)6105 42673 y Fw(\(1976\),)435 b(339{356.)3432 45385 y([14])651 b(T.M.)476 b(Liggett,)486 b Fo(Sto)-66 b(chastic)502 b(inter)-66 b(acting)500 b(systems:)674 b(c)-66 b(ontact,)511 b(voter)502 b(and)h(exclu-)6105 46990 y(sion)465 b(pr)-66 b(o)g(c)g(esses)p Fw(,)433 b(Springer)g(V)-108 b(erlag,)434 b(Berlin,)g(1985.)3432 49703 y([15])651 b(W.)472 b(Magn)-36 b(us,)481 b(F.)472 b(Ob)36 b(erhettinger,)480 b(and)470 b(R.P)-108 b(.)472 b(Soni,)481 b Fo(F)-100 b(ormulas)501 b(and)f(The)-66 b(or)g(ems)6105 51308 y(for)549 b(the)f(Sp)-66 b(e)g(cial)547 b(F)-100 b(unctions)548 b(of)h(Mathematic)-66 b(al)547 b(Physics)p Fw(,)h(Grundlehren)522 b(Band)6105 52913 y(52,)435 b(Springer)d(V)-108 b(erlag,)435 b(Berlin,)f(1966.)3432 55625 y([16])651 b(M.)1064 b(Pr\177)-650 b(ahofer,)1223 b Fo(Sto)-66 b(chastic)1043 b(Surfac)-66 b(e)1043 b(Gr)-66 b(owth)p Fw(,)1222 b(Ph.D.)1065 b(thesis,)1222 b(Ludwig-)6105 57230 y(Maximilians-Univ)-36 b(ersit\177)-650 b(at,)436 b(M)36 b(\177)-686 b(unc)-36 b(hen,)6105 58835 y(h)g(ttp://edo)36 b(c.ub.uni-m)-36 b(uenc)g(hen.de/arc)g(hiv)g(e/00001381,)435 b(2003.)3432 61547 y([17])651 b(M.)312 b(Pr\177)-650 b(ahofer)313 b(and)e(H.)h(Sp)36 b(ohn,)336 b Fo(Curr)-66 b(ent)352 b(\015uctuations)h(for)f(the)h(total)66 b(ly)354 b(asymmet-)6105 63152 y(ric)446 b(simple)g(exclusion)g(pr)-66 b(o)g(c)g(ess)p Fw(,)417 b(In)d(and)f(out)g(of)i(equilibrium)e(\(V.)h (Sidora)-36 b(vicius,)6105 64757 y(ed.\),)434 b(Progress)g(in)f (Probabilit)-36 b(y)-108 b(,)434 b(Birkh\177)-650 b(auser,)434 b(2002.)3432 67469 y([18])651 b(M.)375 b(Pr\177)-650 b(ahofer)376 b(and)e(H.)i(Sp)36 b(ohn,)386 b Fo(Sc)-66 b(ale)410 b(invarianc)-66 b(e)408 b(of)j(the)g(PNG)g(dr)-66 b(oplet)411 b(and)g(the)6105 69074 y(Airy)464 b(pr)-66 b(o)g(c)g(ess)p Fw(,)433 b(J.)h(Stat.)g(Ph)-36 b(ys.)433 b Fh(108)i Fw(\(2002\),)f(1071{1106.)24367 75163 y(51)p eop end end %%Page: 52 52 TeXDict begin HPSdict begin 52 51 bop 3432 7638 a Fw([19])651 b(M.)456 b(Pr\177)-650 b(ahofer)457 b(and)e(H.)i(Sp)36 b(ohn,)461 b Fo(Exact)485 b(sc)-66 b(aling)485 b(function)f(for)h (one-dimensional)6105 9243 y(stationary)464 b(KPZ)h(gr)-66 b(owth)p Fw(,)434 b(J.)g(Stat.)f(Ph)-36 b(ys.)434 b Fh(115)g Fw(\(2004\),)h(255{279.)3432 11955 y([20])651 b(H.)440 b(Sp)36 b(ohn,)441 b Fo(Exc)-66 b(ess)470 b(noise)g(for)h(a)g(lattic) -66 b(e)469 b(gas)i(mo)-66 b(del)470 b(of)g(a)h(r)-66 b(esistor)p Fw(,)441 b(Z.)f(Ph)-36 b(ys.)440 b(B)6105 13560 y Fh(57)434 b Fw(\(1984\),)h(255{261.)3432 16272 y([21])651 b(C.A.)643 b(T)-108 b(racy)642 b(and)f(H.)i(Widom,)694 b Fo(L)-66 b(evel-sp)g(acing)654 b(distributions)h(and)h(the)g(Airy) 6105 17877 y(kernel)p Fw(,)433 b(Comm.)h(Math.)g(Ph)-36 b(ys.)433 b Fh(159)i Fw(\(1994\),)f(151{174.)24367 75163 y(52)p eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0504130757432--