Content-Type: multipart/mixed; boundary="-------------0407261047922" This is a multi-part message in MIME format. ---------------0407261047922 Content-Type: text/plain; name="04-229.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-229.keywords" random walk in random environment ---------------0407261047922 Content-Type: application/postscript; name="mott.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mott.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: mott.dvi %%Pages: 48 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR10 CMBX10 MSBM10 CMMI8 CMMI10 CMSY10 %%+ CMBX12 CMMI12 CMR8 CMTI12 CMMI6 CMSY8 CMR6 CMEX10 CMSL12 CMSY6 MSBM7 %%+ TeX-cmex8 CMBX8 LASY10 CMTT12 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips mott %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.07.21:1830 %%BeginProcSet: texc.pro %! 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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % 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 (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 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 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.001 %%CreationDate: 1992 Oct 23 20:19:17 %%RevisionDate: 2001 Jun 05 20:19:17 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 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 50 /a50 put readonly def /FontBBox{-19 -192 944 683}readonly def /UniqueID 5011949 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % 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 (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 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 /.notdef put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueID 5000766 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 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 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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0.05 %%EndComments 13 dict dup begin /FontInfo 16 dict dup begin /Copyright (see\040copyright\040of\040original\040TeX\040font) def /FamilyName (TeX\040cmex8) def /FullName (TeX\040cmex8\040Regular) def /ItalicAngle 0 def /Notice (converted\040after\040April\0402001) def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) def /isFixedPitch false def /version (001.001) def end readonly def /FontName /TeX-cmex8 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 80 /P put dup 81 /Q put readonly def /FontBBox {-29 -2957 1554 772} readonly def /UniqueID 4314416 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BAE4F093966CD2192CE95EB0F323A6BABFDACCFCF27D91F7869A0E46CA5 9AAF6905783E8AC1F3F9875A76F97187738432F8D14E61574CB292FFB9740871 66839799D8CAF6E0DFE00012EE6D46A2B3655F29705BE37FD5EDA1C765AA2CF5 C5AD37207ED1EE9DB82FF31A33307FFA16911406557336AF92F50B603C7BD336 73EC060F68318378A6F599DDADA5A21504CADBA1E1F4B1A22962BA1BB39ADC7B E8CC92F196549457877C9636A8A7EFAC1C3745644C0FD151C70B9FAD69B02C1F FE5ED071CA1CF3D4A70909B6A3986687D8FAD2D91A9ED24A74DD3F1FE9EA0B50 D076040D6B144A72A89636F74C32D8EADD4DFBB5CD4844DE4D6818A736438AA7 657B523D81D271CF0C94430F8268D4AAEA69261D7E8A3778A2C9BBE16D839984 8157032AE38230F9DE2484D0F0CC96DD3E4C2963466334F685D5F8F1468CFC23 915D2F3759DDE3825A635A29CE151E0AADC574423CC947A0D5F485B227CCE41C 9B7D99BEBB3D10A51E25DEC45074BE221A066734EEB07A150D715353E4AAD204 8CEC24861DDB44C7D4CF4AEC464D0E3D32E3D358608D6B4E5329E4511F9EEFBA 9CD16DAE6A75D65A7E8F58E92790E82C4B14C738ACC86A19E3AEC867FE1EA06A 75D3E1F3ED8744582EBF3E35F02E07F72C9FD51C6251265A4CE916D35D034F1A A5CDCE0ABD996147461504EA4DAE01E34624D1E58958E4928D006C9E1E2903A7 E5BA2F396CC231882119CB3BBAD5AD2B5895BF172B653B83324DF7DE04E9B135 E7B47A831DE5DDA3662991D22EDABA932D04141D6137DBB698141CDCA0538302 3BAEE93934748D65C75A34749C7D7A4D9B0268FEEE83C1EA9D80EE942A955DB3 F9889D0C76F78435A2040699D923FD1D7C91C47F55FD449E0EF716BB8D310EAB 82CA675D5ED19C29C5BB254E27BDB79D4115975666335B87713DCFEA85F78855 549320F2F5A6AB955118F5BF9141F5DDE027D9CD6D484606C64594520C5E9795 8D17B1046CA34CCB9DFA013E4F04E792A022564A431A47E1CA5AD16C4D5AB66A EC3DA6E6C788EA3D471012B48512701FF915D0DA63E58D96291FCD5496CB9851 F1B7CAEF9DAD5582A795DE8ABE5F903B1B1C81711662F04540DDA8E4A8D6BD60 9E809D57DC361134EE1263490A98D7BEE9E66BA8C2B7B8FDBCF14934A05C6993 4FE7D6D88E1FFE0ADB45F0CFFB5772E594162A4740BC1D08E083BA4D4F9A9C10 2DB274D9C154EBF5DA940A56BD6BC5C73521592915E24308F60AD7B7E5AC6FCF 9006DAC20EC91C1F6A3A359F12791A04F68B46E451798E8355FEAC1CA1FB1BD2 0181BCA2E009A8E68EFE4C212306C8E88E84558A15F17F0615FE5D178929BD19 FE2B8EC8BA745853BDF90234B2F2D3A6B9FFFA6C310F257AAF705F4E28323A37 5629D1ABB76061497BE69263677EA993309A16244E680903AE2FB4D65419FC10 9E888B164478BFE62EA5A00870E59610154FD63573391FE49035452F6E7D1297 1DB76688D8ED8431D92A5C7EA2A9D4B1C05FEBFAC333A77120CECD4841E12282 8DCE13DB35B757A78B908B630A063BD10502E9C171D9E936A31BEC9B4B3D9481 31508EC436C871501D95B0C1B0157488CD1D5E2C93E8AC1DE1286C1E69F61078 33FEE4F9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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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 0 /.notdef put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 100 /hatwidest put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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. 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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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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. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 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 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec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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 /.notdef put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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0000000000000000000000000000000000000000000000000000000000000000 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. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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b(\(2\))g(b)m(y)h(maximizing)g(these)h(rates)e(for)g(a)g (\014xed)h(temp)s(erature)g(under)g(the)g(constrain)m(t)g(that)f(the)h (mean)-24 2499 y(densit)m(y)25 b(of)c(p)s(oin)m(ts)i Fv(x)741 2514 y FB(j)800 2499 y Fx(with)g(energies)h(in)e([)p Fw(\000)p Fv(E)6 b Fx(\()p Fv(\014)g Fx(\))p Fv(;)17 b(E)6 b Fx(\()p Fv(\014)g Fx(\)])22 b(is)h(equal)g(to)f Fv(\032)17 b(\027)6 b Fx(\([)p Fw(\000)p Fv(E)g Fx(\()p Fv(\014)g Fx(\))p Fv(;)17 b(E)6 b Fx(\()p Fv(\014)g Fx(\)]\))27 b Fw(\030)h Fv(c)3521 2514 y Ft(0)3561 2499 y Fv(\032E)6 b Fx(\()p Fv(\014)g Fx(\))3826 2463 y Ft(1+)p FB(\013)3965 2499 y Fx(.)-24 2619 y(As)31 b(the)g(c)m(haracteristic)h(mean)f (distance)h Fw(j)p Fv(x)1600 2634 y FB(j)1654 2619 y Fw(\000)18 b Fv(x)1804 2634 y FB(k)1847 2619 y Fw(j)30 b Fx(b)s(et)m(w)m(een)i(sites)g(with)f(optimal)f(jump)h(rates)g(also)f (v)-5 b(aries)-24 2740 y(hea)m(vily)26 b(with)f(the)g(in)m(v)m(erse)i (temp)s(erature)e Fv(\014)6 b Fx(,)26 b(one)e(sp)s(eaks)i(of)e(a)g Fs(variable)i(r)-5 b(ange)27 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b(us)g(discuss)h(the)f(remaining)g(asp)s(ects)h(of)e(the)h(mo) s(del.)-24 1013 y(The)36 b(stationarit)m(y)f(of)g(the)g(underlying)h (simple)g(p)s(oin)m(t)f(pro)s(cess)h Fw(f)p Fv(x)2488 1028 y FB(j)2524 1013 y Fw(g)f Fx(simply)h(re\015ects)g(that)f(the)g (material)-24 1134 y(is)g(homogeneous,)i(while)e(the)g(indep)s(endence) i(of)d(the)h(energy)h(marks)f(is)g(compatible)g(with)h(P)m(oisson)f (lev)m(el)-24 1254 y(statistics,)41 b(whic)m(h)e(is)f(a)f(general)h (rough)f(indicator)h(for)f(the)h(lo)s(calization)g(regime)g(and)f(has)h (b)s(een)h(pro)m(v)m(en)-24 1374 y(to)h(hold)g(for)g(an)g(Anderson)h (mo)s(del)g([Min].)66 b(The)41 b(exp)s(onen)m(t)h Fv(\013)f Fx(allo)m(ws)g(to)e(mo)s(del)i(a)f(p)s(ossible)h(Coulom)m(b)-24 1495 y(pseudogap)34 b(in)f(the)g(densit)m(y)h(of)e(states)i([SE].)123 1662 y(Ha)m(ving)e(in)g(mind)g(the)g(Einstein)h(relation)e(b)s(et)m(w)m (een)j(the)e(conductivit)m(y)i(and)d(the)h(di\013usion)g(co)s (e\016cien)m(t)-24 1783 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b(criterion)f(then)h (follo)m(ws)f(b)m(y)h(expanding)g(the)f Fv(k)s Fx(-th)f(p)s(o)m(w)m(er) i(of)e(the)h(ab)s(o)m(v)m(e)g(lo)m(w)m(er)h(and)f(upp)s(er)g(b)s (ounds,)-24 5315 y(applying)h(Lemma)f(1\(iv\))g(and)f(using)i(the)f (stationarit)m(y)g(of)f Fw(P)8 b Fx(.)1605 b Ff(2)1935 5564 y Fx(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 123 172 a Fx(Let)29 b(us)g(p)s(oin)m(t)h(out)e (that)h Fv(c)1086 187 y FB(x;y)1187 172 y Fx(\()p Fv(\030)5 b Fx(\))28 b(and)h Fv(\025)1582 187 y FB(x)1626 172 y Fx(\()p Fv(\030)5 b Fx(\))28 b(are)h Fs(c)-5 b(ovariant)p Fx(,)29 b(that)g(is,)h(for)e(an)m(y)i Fv(\030)i Fw(2)c(N)3293 187 y Ft(0)3361 172 y Fx(and)h Fv(y)t(;)17 b(z)32 b Fw(2)3817 146 y Fo(d)3814 172 y Fv(S)3874 187 y FB(x)3918 172 y Fv(\030)t Fx(,)884 413 y Fv(c)926 428 y FB(z)s(;y)1022 413 y Fx(\()p Fv(S)1120 428 y FB(x)1164 413 y Fv(\030)5 b Fx(\))55 b(=)g Fv(c)1478 428 y FB(z)s Ft(+)p FB(x;y)r Ft(+)p FB(x)1764 413 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fv(;)212 b(\025)2211 428 y FB(y)2252 413 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b(\(C1\))g(and)g(\(C2\).)123 1162 y Fx(The)k (construction)g(of)f(the)h(dynamics)h(follo)m(ws)f(standard)f (references)j(\()p Fn(e.g.)57 b Fx([Bre)q(,)38 b(Chapter)g(15])f(and) -24 1282 y([Kal,)k(Chapter)e(12]\).)62 b(After)38 b(arriving)h(at)g (site)g Fv(y)j Fw(2)2007 1256 y Fx(^)1997 1282 y Fv(\030)5 b Fx(,)40 b(the)f(particle)h(w)m(aits)f(an)g(exp)s(onen)m(tial)h(time)f (with)-24 1402 y(parameter)33 b Fv(\025)499 1417 y FB(y)541 1402 y Fx(\()p Fv(\030)5 b Fx(\))32 b(and)g(then)h(jumps)h(to)e (another)h(site)g Fv(z)g Fw(2)2237 1376 y Fx(^)2227 1402 y Fv(\030)k Fx(with)c(probabilit)m(y)1539 1673 y Fv(p)1588 1632 y FB(\030)1626 1673 y Fx(\()p Fv(z)t Fw(j)p Fv(y)t Fx(\))55 b(:=)2054 1606 y Fv(c)2096 1621 y FB(y)r(;z)2193 1606 y Fx(\()p Fv(\030)5 b Fx(\))p 2054 1650 262 4 v 2074 1741 a Fv(\025)2131 1756 y FB(y)2173 1741 y Fx(\()p Fv(\030)g Fx(\))2359 1673 y(.)1433 b(\(15\))-24 1927 y(More)47 b(precisely)-8 b(,)52 b(consider)c Fv(\030)56 b Fw(2)c(N)1370 1942 y Ft(0)1455 1927 y 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b(fr)-5 b(om)38 b(the)h(p)-5 b(article)44 b Fx(p)s(erforming)36 b(the)h(discrete{time)i (random)-24 5037 y(w)m(alk)27 b(with)e(distribution)950 5012 y(~)936 5037 y Fm(P)1013 4990 y FB(\030)1013 5062 y Ft(0)1052 5037 y Fx(.)41 b(Let)25 b(us)h(p)s(oin)m(t)f(out)g(a)g(few) h(prop)s(erties)g(of)f(the)h(distribution)3315 5012 y(~)3301 5037 y Fm(P)3378 5052 y FB(\030)3415 5037 y Fx(.)41 b(First,)27 b(w)m(e)f(re-)-24 5167 y(mark)i(that)g(due)g(to)g(\(14\),)g(the)g(pro)s (cess)1451 5086 y Fo(\000)1496 5167 y Fv(\025)1573 5183 y Ft(~)1553 5200 y FB(X)1616 5167 y Fr(\030)1611 5210 y(n)1658 5167 y Fx(\()p Fv(\030)5 b Fx(\))1782 5086 y Fo(\001)1827 5206 y FB(n)p Fq(2)p Fj(N)1997 5167 y Fx(de\014ned)29 b(on)e(\()2507 5142 y(~)2496 5167 y(\012)2566 5182 y FB(\030)2604 5167 y Fv(;)2662 5141 y Fx(~)2648 5167 y Fm(P)2725 5120 y FB(\030)2725 5191 y Ft(0)2764 5167 y Fx(\))h(and)f(the)i(pro)s(cess)f(\()p Fv(\025)3607 5182 y Ft(0)3647 5167 y Fx(\()p Fv(\030)3728 5182 y FB(n)3774 5167 y Fx(\)\))3850 5182 y FB(n)p Fq(2)p Fj(N)1935 5564 y Fx(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop -24 171 a Fx(de\014ned)34 b(on)d(\()492 145 y(~)484 171 y(\004)p Fv(;)607 145 y Fx(~)593 171 y Fm(P)670 186 y FB(\030)707 171 y Fx(\))h(ha)m(v)m(e)h(the)f(same)h (distribution.)44 b(Moreo)m(v)m(er,)34 b(due)e(to)g(Remark)g(1,)g(if)f Fv(\030)i Fw(2)28 b(W)8 b Fx(,)32 b(then)h(the)-24 291 y(pro)s(cess)h(\()p Fv(\020)397 306 y FB(n)444 291 y Fx(\))482 306 y FB(n)p Fq(2)p Fj(N)656 291 y Fx(de\014ned)g(on)f(\() 1174 266 y(~)1166 291 y(\004)p Fv(;)1288 266 y Fx(~)1275 291 y Fm(P)1352 306 y FB(\030)1389 291 y Fx(\))f(as)1148 594 y Fv(\020)1191 609 y FB(n)1293 594 y Fx(=)1424 424 y Fo(\()1504 526 y Fx(0)802 b(if)32 b Fv(n)c Fx(=)f(0)17 b Fv(;)1504 595 y Fo(P)1609 622 y FB(n)p Fq(\000)p Ft(1)1609 699 y FB(k)r Ft(=0)1763 670 y Fx(\001\()p Fv(\030)1925 685 y FB(k)1967 670 y Fv(;)g(\030)2054 685 y FB(k)r Ft(+1)2187 670 y Fx(\))130 b(otherwise)29 b Fv(;)-24 874 y Fx(has)j(paths)g(in)537 849 y(~)526 874 y(\012)596 889 y FB(\030)666 874 y Fx(with)f (distribution)1431 849 y(~)1417 874 y Fm(P)1494 827 y FB(\030)1494 899 y Ft(0)1533 874 y Fx(.)43 b(Finally)-8 b(,)32 b(it)f(is)h(con)m(v)m(enien)m(t)i(to)c(consider)j(a)e(suitable)h (a)m(v)m(erage)h(of)-24 995 y(the)g(distributions)728 969 y(~)715 995 y Fm(P)792 1010 y FB(\030)829 995 y Fx(.)43 b(T)-8 b(o)33 b(this)g(aim,)g(let)g Fw(Q)1672 1010 y Ft(0)1744 995 y Fx(b)s(e)g(the)g(probabilit)m(y)g(measure)h(on)f Fw(N)3141 1010 y Ft(0)3213 995 y Fx(de\014ned)h(as)1371 1265 y Fw(Q)1452 1280 y Ft(0)1491 1265 y Fx(\()p Fv(d\030)5 b Fx(\))55 b(:=)1946 1198 y Fv(\025)2003 1213 y Ft(0)2042 1198 y Fx(\()p Fv(\030)5 b Fx(\))p 1889 1242 334 4 v 1889 1334 a Fm(E)1963 1349 y Fq(P)2012 1358 y Fp(0)2051 1334 y Fx(\()p Fv(\025)2146 1349 y Ft(0)2185 1334 y Fx(\))2260 1265 y Fw(P)2329 1280 y Ft(0)2369 1265 y Fx(\()p Fv(d\030)g Fx(\))27 b Fv(;)-24 1512 y Fx(and)33 b(set)332 1487 y(~)318 1512 y Fm(P)28 b Fx(:=)553 1432 y Fo(R)636 1512 y Fw(Q)717 1527 y Ft(0)757 1512 y Fx(\()p Fv(d\030)5 b Fx(\))946 1487 y(~)932 1512 y Fm(P)1009 1527 y FB(\030)1045 1512 y Fx(.)44 b(Note)33 b(that,)f(if)g Fv(\030)g Fw(2)d(W)8 b Fx(,)33 b(the)g(transition)g(probabilities)g(are)564 1815 y Fv(p)p Fx(\()p Fv(\030)699 1774 y Fq(0)722 1815 y Fw(j)p Fv(\030)5 b Fx(\))54 b(:=)1063 1790 y(~)1049 1815 y Fm(P)1125 1735 y Fo(\000)1171 1815 y Fv(\030)1214 1830 y FB(n)p Ft(+1)1378 1815 y Fx(=)28 b Fv(\030)1530 1774 y Fq(0)1569 1815 y Fw(j)p Fv(\030)1640 1830 y FB(n)1714 1815 y Fx(=)g Fv(\030)1866 1735 y Fo(\001)1966 1815 y Fx(=)2097 1645 y Fo(\()2178 1747 y Fv(\025)2235 1706 y Fq(\000)p Ft(1)2235 1772 y(0)2329 1747 y Fx(\()p Fv(\030)5 b Fx(\))p Fv(c)2495 1762 y Ft(0)p FB(;x)2593 1747 y Fx(\()p Fv(\030)g Fx(\))180 b(if)32 b Fv(\030)3034 1711 y Fq(0)3085 1747 y Fx(=)27 b Fv(S)3248 1762 y FB(x)3292 1747 y Fv(\030)32 b(;)2178 1892 y Fx(0)670 b(otherwise)34 b(.)-24 2140 y Fm(Lemma)39 b(3)49 b Fs(The)g(discr)-5 b(ete-time)48 b(Markov)h(pr)-5 b(o)g(c)g(ess)48 b Fx(\()p Fv(\030)2077 2155 y FB(n)2124 2140 y Fx(\))2162 2155 y FB(n)p Fq(\025)p Ft(0)2348 2140 y Fs(de\014ne)-5 b(d)48 b(on)h Fx(\()2891 2114 y(~)2883 2140 y(\004)p Fv(;)3005 2114 y Fx(~)2992 2140 y Fm(P)o Fx(\))g Fs(is)g(r)-5 b(eversible)48 b(and)g(is)-24 2260 y(\(time\))35 b(er)-5 b(go)g(dic)34 b(if)g Fw(P)44 b Fs(is)34 b(er)-5 b(go)g(dic.)-24 2447 y Fm(Pro)s(of.)50 b Fx(Rev)m(ersibilit)m(y)36 b(means)d(that)178 2574 y Fo(Z)233 2800 y Fq(N)291 2809 y Fp(0)346 2710 y Fw(Q)427 2725 y Ft(0)467 2710 y Fx(\()p Fv(d\030)5 b Fx(\))659 2574 y Fo(Z)784 2683 y Fx(^)774 2710 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))17 b Fv(p)p Fx(\()p Fv(S)1167 2725 y FB(x)1210 2710 y Fv(\030)5 b Fw(j)p Fv(\030)g Fx(\))17 b Fv(f)11 b Fx(\()p Fv(\030)5 b Fx(\))17 b Fv(g)t Fx(\()p Fv(S)1738 2725 y FB(x)1778 2710 y Fv(\030)5 b Fx(\))55 b(=)2050 2574 y Fo(Z)2105 2800 y Fq(N)2163 2809 y Fp(0)2219 2710 y Fw(Q)2300 2725 y Ft(0)2339 2710 y Fx(\()p Fv(d\030)5 b Fx(\))2531 2574 y Fo(Z)2656 2683 y Fx(^)2646 2710 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))17 b Fv(p)p Fx(\()p Fv(S)3040 2725 y FB(x)3083 2710 y Fv(\030)5 b Fw(j)p Fv(\030)g Fx(\))17 b Fv(g)t Fx(\()p Fv(\030)5 b Fx(\))17 b Fv(f)11 b Fx(\()p Fv(S)3611 2725 y FB(x)3651 2710 y Fv(\030)5 b Fx(\))27 b Fv(;)-24 2991 y Fx(for)47 b(an)m(y)g Fv(f)5 b(;)17 b(g)56 b Fw(2)d(F)10 b Fx(\()p Fw(N)859 3006 y Ft(0)897 2991 y Fx(\).)87 b(The)48 b(ab)s(o)m(v)m(e)g(iden)m(tit)m(y)g (can)g(b)s(e)f(inferred)h(from)e(Lemma)i(1\(i\))f(b)m(y)h(means)g(of) -24 3112 y(straigh)m(tforw)m(ard)31 b(computation)f(using)g(the)g(co)m (v)-5 b(ariance)31 b(relations)f(\(14\))f(and)h(the)g(symmetry)h(of)f (the)g(rates)-24 3232 y Fv(c)18 3247 y FB(x;y)119 3232 y Fx(\()p Fv(\030)5 b Fx(\).)123 3376 y(Due)34 b(to)h(Corollary)g(5)f (in)h([Ros,)g(Chapter)h(IV],)f(in)g(order)g(to)f(pro)m(v)m(e)i(ergo)s (dicit)m(y)g(it)e(is)h(enough)h(to)e(sho)m(w)-24 3497 y(that)48 b Fw(Q)284 3512 y Ft(0)323 3497 y Fx(\()p Fv(A)p Fx(\))53 b Fw(2)g(f)p Fx(0)p Fv(;)17 b Fx(1)p Fw(g)47 b Fx(if)g Fv(A)53 b Fw(2)g(B)s Fx(\()p Fw(N)1470 3512 y Ft(0)1510 3497 y Fx(\))47 b(has)h(the)f(follo)m(wing)h(prop)s(ert)m (y:)74 b(for)47 b Fw(Q)3104 3512 y Ft(0)3144 3497 y Fx({almost)g(all)g Fv(\030)58 b Fw(2)53 b Fv(A)p Fx(,)-10 3591 y(~)-24 3617 y Fm(P)p Fx(\()p Fv(\030)134 3632 y Ft(1)218 3617 y Fw(2)46 b Fv(A)p Fw(j)p Fv(\030)474 3632 y Ft(0)558 3617 y Fx(=)f Fv(\030)5 b Fx(\))45 b(=)g Fv(\037)992 3632 y FB(A)1049 3617 y Fx(\()p Fv(\030)5 b Fx(\).)74 b(Note)43 b(that)g(this)h(prop)s (ert)m(y)f(implies)i(for)d Fw(Q)2933 3632 y Ft(0)2973 3617 y Fx({almost)h(all)g Fv(\030)50 b Fw(2)45 b Fv(A)e Fx(that)-24 3737 y Fv(\030)24 3701 y Fq(0)82 3737 y Fw(2)36 b Fv(A)h Fx(if)g Fv(p)p Fx(\()p Fv(\030)523 3701 y Fq(0)546 3737 y Fw(j)p Fv(\030)5 b Fx(\))34 b Fv(>)i Fx(0.)56 b(Since)39 b Fw(Q)1279 3752 y Ft(0)1355 3737 y Fx(and)f Fw(P)1619 3752 y Ft(0)1695 3737 y Fx(are)f(absolutely)i(con)m(tin)m (uous)g(w.r.t.)57 b(eac)m(h)38 b(other)f(and)h(since)-24 3858 y Fv(p)p Fx(\()p Fv(S)123 3873 y FB(x)167 3858 y Fv(\030)5 b Fw(j)p Fv(\030)g Fx(\))26 b Fv(>)i Fx(0)k(for)g(all)h Fv(x)28 b Fw(2)1012 3831 y Fx(^)1002 3858 y Fv(\030)t Fx(,)33 b(the)g(statemen)m(t)h(follo)m(ws)f(from)g(Lemma)g(1\(iii\).) 1056 b Ff(2)-24 4025 y Fm(Pro)s(of)42 b(of)f(Prop)s(osition)h(1)p Fx(.)54 b(The)37 b(uniqueness)i(follo)m(ws)e(from)f([Bre)q(,)h(Chapter) f(15].)54 b(In)37 b(order)f(to)g(pro)m(v)m(e)-24 4146 y(existence,)41 b(due)d(to)e(the)i(ab)s(o)m(v)m(e)g(construction,)h(w)m (e)f(only)f(need)h(to)f(pro)m(v)m(e)h(\(16\))f(for)f Fw(P)3203 4161 y Ft(0)3243 4146 y Fx({almost)h(all)g Fv(\030)k Fx(and)-24 4266 y(for)32 b(an)m(y)i Fv(x)28 b Fw(2)497 4240 y Fx(^)487 4266 y Fv(\030)t Fx(.)44 b(According)33 b(to)f([Bre)q(,)g(Prop.)44 b(15.43],)32 b(condition)h(\(16\))f(is)h (implied)h(b)m(y)g(the)f(follo)m(wing)f(one:)1369 4532 y(~)1355 4558 y Fm(P)1432 4517 y FB(\030)1432 4583 y(x)1475 4447 y Fo(\020)1588 4433 y Fq(1)1551 4463 y Fo(X)1557 4672 y FB(n)p Ft(=0)1840 4490 y Fx(1)p 1722 4535 285 4 v 1722 4626 a Fv(\025)1799 4642 y Ft(~)1779 4659 y FB(X)1842 4626 y Fr(\030)1837 4670 y(n)1883 4626 y Fx(\()p Fv(\030)5 b Fx(\))2044 4558 y(=)28 b Fw(1)2265 4447 y Fo(\021)2379 4558 y Fx(=)55 b(1)28 b Fv(:)1205 b Fx(\(17\))-24 4814 y(Due)33 b(to)f(the)h(iden)m(tit)m(y)404 5080 y(~)390 5106 y Fm(P)467 5058 y FB(\030)467 5130 y Ft(0)506 4995 y Fo(\020)619 4981 y Fq(1)582 5011 y Fo(X)588 5220 y FB(n)p Ft(=1)871 5038 y Fx(1)p 753 5083 V 753 5174 a Fv(\025)830 5190 y Ft(~)810 5207 y FB(X)873 5174 y Fr(\030)868 5217 y(n)915 5174 y Fx(\()p Fv(\030)5 b Fx(\))1075 5106 y(=)28 b Fw(1)1306 5021 y Fo(\014)1306 5081 y(\014)1395 5080 y Fx(~)1367 5106 y Fv(X)1456 5058 y FB(\030)1448 5130 y Ft(1)1521 5106 y Fx(=)g Fv(x)1697 4995 y Fo(\021)1812 5106 y Fx(=)1957 5080 y(~)1943 5106 y Fm(P)2020 5064 y FB(\030)2020 5130 y(x)2064 4995 y Fo(\020)2176 4981 y Fq(1)2140 5011 y Fo(X)2145 5220 y FB(n)p Ft(=0)2428 5038 y Fx(1)p 2310 5083 V 2310 5174 a Fv(\025)2387 5190 y Ft(~)2367 5207 y FB(X)2430 5174 y Fr(\030)2425 5217 y(n)2472 5174 y Fx(\()p Fv(\030)5 b Fx(\))2633 5106 y(=)27 b Fw(1)2853 4995 y Fo(\021)2940 5106 y Fv(;)211 b Fw(8)28 b Fv(x)g Fw(2)3449 5079 y Fx(^)3439 5106 y Fv(\030)37 b Fx(,)1935 5564 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop -24 171 a Fx(the)27 b(pro)s(of)e(will)h(b)s(e)g (completed)i(if)d(w)m(e)i(can)g(sho)m(w)g(\(17\))e(for)g Fv(x)j Fx(=)g(0)d(and)h Fw(P)2630 186 y Ft(0)2670 171 y Fx({almost)g(all)g Fv(\030)k Fx(and,)e(in)e(particular,)-24 291 y(if)33 b(w)m(e)g(can)g(sho)m(w)665 557 y(~)652 583 y Fm(P)728 502 y Fo(\000)810 458 y Fq(1)774 488 y Fo(X)779 697 y FB(n)p Ft(=0)1051 515 y Fx(1)p 944 560 262 4 v 944 651 a Fv(\025)1001 666 y Ft(0)1041 651 y Fx(\()p Fv(\030)1122 666 y FB(n)1168 651 y Fx(\))1244 583 y(=)27 b Fw(1)1447 502 y Fo(\001)1548 583 y Fx(=)1679 447 y Fo(Z)1795 583 y Fw(Q)1876 598 y Ft(0)1916 583 y Fx(\()p Fv(d\030)5 b Fx(\))2132 557 y(~)2118 583 y Fm(P)2195 536 y FB(\030)2195 607 y Ft(0)2234 472 y Fo(\020)2347 458 y Fq(1)2310 488 y Fo(X)2315 697 y FB(n)p Ft(=0)2598 515 y Fx(1)p 2480 560 285 4 v 2480 651 a Fv(\025)2557 667 y Ft(~)2537 684 y FB(X)2600 651 y Fr(\030)2595 694 y(n)2642 651 y Fx(\()p Fv(\030)g Fx(\))2803 583 y(=)27 b Fw(1)3023 472 y Fo(\021)3110 583 y Fx(=)g(1)h Fv(;)-24 812 y Fx(or)651 1051 y(~)637 1077 y Fm(P)714 996 y Fo(\000)783 1077 y Fx(lim)759 1137 y FB(n)p Fq(!1)960 1077 y Fv(\025)1017 1092 y Ft(0)1056 1077 y Fx(\()p Fv(\030)1137 1092 y FB(n)1184 1077 y Fx(\))f(=)h Fw(1)1453 996 y Fo(\001)1553 1077 y Fx(=)1685 941 y Fo(Z)1801 1077 y Fw(Q)1882 1092 y Ft(0)1922 1077 y Fx(\()p Fv(d\030)5 b Fx(\))2137 1051 y(~)2123 1077 y Fm(P)2200 1029 y FB(\030)2200 1101 y Ft(0)2239 966 y Fo(\020)2340 1077 y Fx(lim)2315 1137 y FB(n)p Fq(!1)2516 1077 y Fv(\025)2593 1093 y Ft(~)2573 1110 y FB(X)2636 1077 y Fr(\030)2631 1120 y(n)2678 1077 y Fx(\()p Fv(\030)g Fx(\))27 b(=)g Fw(1)3049 966 y Fo(\021)3135 1077 y Fx(=)h(0)17 b Fv(:)123 1316 y Fx(If)30 b Fw(P)39 b Fx(is)31 b(ergo)s(dic,)g(then)g (Lemma)g(3)g(implies)h(that)1984 1291 y(~)1970 1316 y Fm(P)e 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b(This)27 b(induces)g(a)f(map)g Fv(S)1721 3346 y FB(T)1801 3331 y Fx(on)g Fw(N)40 b Fx(b)m(y)27 b(\()p Fv(S)2279 3346 y FB(T)2334 3331 y Fv(\030)5 b Fx(\)\()p Fv(B)g Fx(\))27 b(=)g Fv(\030)5 b Fx(\()p Fv(S)2851 3346 y FB(T)2906 3331 y Fv(B)g Fx(\))25 b(where)i Fv(B)33 b Fw(2)28 b(B)s Fx(\()p Fu(R)3702 3295 y FB(d)3751 3331 y Fw(\002)8 b Fv(K)f Fx(\).)-24 3452 y(Then,)30 b(as)d(in)g([DF)m(GW,)h (p.)42 b(824],)27 b(one)g(can)h(pro)m(v)m(e)g(that)f(if)g Fw(P)35 b Fx(is)28 b(re\015ection)g(in)m(v)-5 b(arian)m(t)27 b(\()p Fs(i.e.)41 b Fw(P)20 b(\016)11 b Fv(S)3557 3467 y FB(R)3610 3477 y Fr(i)3668 3452 y Fx(=)27 b Fw(P)36 b Fx(for)-24 3572 y(all)e(re\015ections)g Fv(R)648 3587 y FB(i)710 3572 y Fx(with)g Fv(R)1007 3587 y FB(i)1035 3572 y Fv(x)29 b Fx(=)g(\()p Fv(x)1317 3536 y Ft(\(1\))1412 3572 y Fv(;)17 b(:)g(:)g(:)f(;)h Fw(\000)p Fv(x)1763 3536 y Ft(\()p FB(i)p Ft(\))1846 3572 y Fv(;)g(:)g(:)g(:)f(;)h(x)2120 3536 y Ft(\()p FB(d)p Ft(\))2216 3572 y Fx(\)\),)33 b(then)h(the)f 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b(this)h(end,)i(let)e(us)g(\014rst)g(mak)m(e)h(a)e(general)h(observ)-5 b(ation.)42 b(Let)26 b Fv(\025)i(>)g Fx(0)e(and)g(let)h Fv(T)3115 858 y Ft(1)3155 843 y Fv(;)17 b(:)g(:)g(:)f(;)h(T)3431 858 y FB(k)3500 843 y Fx(b)s(e)26 b(indep)s(en-)-24 963 y(den)m(t)39 b(exp)s(onen)m(tial)f(v)-5 b(ariables)38 b(on)f(some)h(probabilit)m(y)h(space)f(\(\012)p Fv(;)17 b(\026)p Fx(\),)38 b(with)g(parameters)g Fv(\025)3402 978 y Ft(1)3441 963 y Fv(;)17 b(:)g(:)g(:)f(;)h(\025)3717 978 y FB(k)3795 963 y Fw(\024)36 b Fv(\025)p Fx(.)-24 1084 y(De\014ne)j(the)f(random)g(v)-5 b(ariables)39 b Fv(T)1301 1047 y Fq(0)1287 1108 y FB(j)1361 1084 y Fx(:=)e(\()p Fv(\025)1596 1099 y FB(j)1632 1084 y Fv(=\025)p Fx(\))p Fv(T)1833 1099 y FB(j)1869 1084 y Fx(,)j Fv(j)j Fx(=)36 b(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(k)s Fx(.)59 b(These)40 b(are)e(indep)s(enden)m(t)i(iden)m(tically)-24 1204 y(distributed)35 b(exp)s(onen)m(tial)e(v)-5 b(ariables)34 b(with)f(parameter)g Fv(\025)p Fx(.)43 b(As)34 b Fv(T)2434 1168 y Fq(0)2420 1229 y FB(j)2484 1204 y Fw(\024)28 b Fv(T)2646 1219 y FB(j)2683 1204 y Fx(,)33 b(this)g(sho)m(ws)h(that)233 1508 y Fv(\026)292 1428 y Fo(\000)337 1508 y Fv(T)394 1523 y Ft(1)456 1508 y Fx(+)22 b Fw(\001)17 b(\001)g(\001)j Fx(+)i Fv(T)847 1523 y FB(k)918 1508 y Fw(\024)28 b Fv(t)1058 1428 y Fo(\001)1159 1508 y Fw(\024)56 b Fv(\026)1351 1428 y Fo(\000)1396 1508 y Fv(T)1467 1467 y Fq(0)1453 1533 y Ft(1)1515 1508 y Fx(+)22 b Fw(\001)17 b(\001)g(\001)j Fx(+)i Fv(T)1920 1467 y Fq(0)1906 1533 y FB(k)1977 1508 y Fw(\024)28 b Fv(t)2117 1428 y Fo(\001)2218 1508 y Fx(=)56 b Fv(e)2395 1467 y Fq(\000)p FB(\025t)2574 1384 y Fq(1)2537 1414 y Fo(X)2548 1624 y FB(j)t Ft(=0)2720 1441 y Fx(\()p Fv(\025t)p Fx(\))2888 1405 y FB(j)t Ft(+)p FB(k)p 2708 1485 324 4 v 2708 1577 a Fx(\()p Fv(j)28 b Fx(+)22 b Fv(k)s Fx(\)!)3096 1508 y Fw(\024)3239 1441 y Fx(\()p Fv(\025t)p Fx(\))3407 1405 y FB(k)p 3239 1485 211 4 v 3304 1577 a Fv(k)s Fx(!)3492 1508 y(.)300 b(\(30\))123 1819 y(In)33 b(order)f(to)h(pro)s(ceed,)g(w)m(e)h(set)f(for)f(all)g Fv(\030)h Fw(2)28 b(N)1838 1834 y Ft(0)1329 2035 y Fv(B)1408 1994 y FB(\030)1403 2060 y(n)1505 2035 y Fx(:=)1664 1954 y Fo(\010)1722 2035 y Fv(x)g Fw(2)1909 2009 y Fx(^)1899 2035 y Fv(\030)49 b Fx(:)44 b Fv(\025)2119 2050 y FB(x)2163 2035 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fw(\024)h Fv(n)2477 1994 y FB(\013)2527 1954 y Fo(\011)2612 2035 y Fv(;)-24 2251 y Fx(as)33 b(w)m(ell)h(as)595 2371 y Fv(A)668 2330 y FB(\030)668 2396 y(n)771 2371 y Fx(:=)929 2261 y Fo(n)996 2291 y(\000)1069 2346 y Fx(~)1041 2371 y Fv(X)1130 2324 y FB(\030)1122 2399 y(k)1168 2371 y Fx(\))1206 2386 y FB(k)r Fq(\025)p Ft(0)1366 2371 y Fw(2)1471 2346 y Fx(~)1460 2371 y(\012)1530 2386 y FB(\030)1613 2371 y Fx(:)72 b Fw(9)28 b Fv(J)37 b Fw(\032)28 b Fv(I)2034 2386 y FB(n)2097 2371 y Fv(;)34 b Fw(j)p Fv(J)9 b Fw(j)27 b Fv(>)2417 2304 y(n)p 2417 2348 59 4 v 2422 2440 a Fx(2)2502 2371 y Fv(;)2591 2346 y Fx(~)2563 2371 y Fv(X)2652 2324 y FB(\030)2644 2397 y(j)2717 2371 y 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3437 y FB(\030)2900 3533 y(j;)2968 3516 y Ft(~)2949 3533 y FB(X)3012 3500 y Fr(\030)3007 3555 y(j)3077 3484 y Fw(\024)28 b Fv(t)p Fw(g)p Fx(.)44 b(As)33 b(a)f(result,)359 3773 y Fv(g)406 3788 y FB(n)453 3773 y Fx(\()p Fv(\030)5 b Fx(\))82 b Fw(\024)970 3678 y Fo(X)820 3894 y FB(J)6 b Fq(\032)p FB(I)951 3902 y Fr(n)992 3894 y FB(;)p Fq(j)p FB(J)g Fq(j)p FB(>n=)p Ft(2)1443 3678 y Fo(X)1309 3908 y FB(x)1349 3917 y Fp(0)1383 3908 y FB(;:::)n(;x)1521 3917 y Fr(n)p Fk(\000)p Fp(1)1642 3908 y Fq(2)1696 3890 y Ft(^)1689 3908 y FB(\030)1739 3773 y Fv(\037)1800 3692 y Fo(\000)1846 3773 y Fv(x)1901 3788 y FB(j)1965 3773 y Fw(2)28 b Fv(B)2138 3732 y FB(\030)2133 3797 y(n)2225 3773 y Fw(8)g Fv(j)33 b Fw(2)28 b Fv(J)2538 3692 y Fo(\001)2601 3773 y Fv(\037)2662 3692 y Fo(\000)2708 3773 y Fv(x)2763 3788 y FB(i)2830 3773 y Fv(=)-60 b Fw(2)28 b Fv(B)2992 3732 y FB(\030)2987 3797 y(n)3078 3773 y Fw(8)g Fv(i)g Fw(2)g Fv(I)3359 3788 y FB(n)3429 3773 y Fw(n)21 b Fv(J)3563 3692 y Fo(\001)972 4056 y Fx(~)958 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b(giv)m(en)i(in)f(\(13\))f(is)h(w)m(ell)h(de\014ned)g(and)f (its)g(distribution)h(on)f(the)g(path)-24 3270 y(space)45 b(\012)318 3285 y FB(\030)402 3270 y Fx(=)h Fv(D)s Fx(\([0)p Fv(;)17 b Fw(1)p Fx(\))p Fv(;)g Fx(supp\()1196 3244 y(^)1186 3270 y Fv(\030)t Fx(\)\))43 b(has)h(b)s(een)g(denoted)h(b)m(y)f Fm(P)2381 3223 y FB(\030)2381 3294 y Ft(0)2420 3270 y Fx(.)76 b(The)44 b Fs(envir)-5 b(onment)44 b(viewe)-5 b(d)44 b(fr)-5 b(om)44 b(the)-24 3399 y(p)-5 b(article)31 b Fx(is)g(the)g(con)m(tin)m(uous{time)i(jump)f(Mark)m(o)m(v)g(pro)s (cess)2249 3319 y Fo(\000)2295 3399 y Fv(S)2355 3432 y FB(X)2418 3400 y Fr(\030)2413 3452 y(t)2456 3399 y Fv(\030)5 b Fx(\))2542 3414 y FB(t)p Fq(\025)p Ft(0)2692 3399 y Fx(de\014ned)32 b(on)f(\(\012)3268 3414 y FB(\030)3306 3399 y Fv(;)17 b Fm(P)3427 3352 y FB(\030)3427 3424 y Ft(0)3466 3399 y Fx(\).)43 b(Let)31 b Fm(P)3824 3414 y FB(\030)3892 3399 y Fx(b)s(e)-24 3535 y(its)k(distribution)f(on)g (the)g(path)g(space)h(\004)30 b(:=)f Fv(D)s Fx(\([0)p Fv(;)17 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Fo(\000)2169 1782 y Fv(n)2227 1797 y Fq(\003)2266 1782 y Fx(\()p Fv(t)p Fx(\))28 b(=)g Fv(n;)17 b(\030)2654 1797 y FB(R)2707 1806 y Fp(1)2773 1782 y Fx(=)27 b Fv(\030)2924 1741 y Ft(\()p FB(n)p Fq(\000)p Ft(1\))3116 1782 y Fv(;)17 b(:)g(:)g(:)f(;)h(\030)3378 1797 y FB(R)3431 1805 y Fr(n)3504 1782 y Fx(=)28 b Fv(\030)3656 1741 y Ft(\(0\))3750 1702 y Fo(\001)3828 1782 y Fx(.)-24 1998 y(where,)42 b(giv)m(en)d Fv(\030)p 553 2033 48 4 v 42 w Fw(2)f Fx(\004,)i Fv(R)948 2013 y Ft(1)988 1998 y Fx(\()p Fv(\030)p 1026 2033 V 4 w Fx(\))e Fv(<)f(R)1336 2013 y Ft(2)1376 1998 y Fx(\()p Fv(\030)p 1414 2033 V 4 w Fx(\))h Fv(<)f(:)17 b(:)g(:)54 b Fx(denote)40 b(the)e(jump)h(times) h(of)e(the)g(path)h Fv(\030)p 3362 2033 V 4 w Fx(.)61 b(Next,)41 b(giv)m(en)-24 2118 y Fv(f)5 b(;)17 b(g)31 b Fw(2)d(F)10 b Fx(\()p Fw(N)447 2133 y Ft(0)486 2118 y Fx(\))32 b(one)h(gets)g(b)m(y)h(applying)f(Lemma)g(1\(i\))g(and)f (using)i(\(33\))e(that)131 2342 y Fo(Z)247 2477 y Fw(P)316 2492 y Ft(0)356 2477 y Fx(\()p Fv(d\030)5 b Fx(\))548 2342 y Fo(Z)673 2451 y Fx(^)663 2477 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))17 b Fv(p)958 2492 y FB(t)987 2477 y Fx(\()p Fv(S)1085 2492 y FB(x)1129 2477 y Fv(\030)5 b Fw(j)p Fv(\030)g Fx(\))26 b Fv(f)11 b Fx(\()p Fv(\030)5 b Fx(\))p Fv(g)t Fx(\()p Fv(S)1649 2492 y FB(x)1692 2477 y Fv(\030)g Fx(\))54 b(=)1963 2342 y Fo(Z)2080 2477 y Fw(P)2149 2492 y Ft(0)2188 2477 y Fx(\()p Fv(d\030)5 b Fx(\))2380 2342 y Fo(Z)2505 2451 y Fx(^)2495 2477 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))17 b Fv(p)2791 2492 y FB(t)2820 2477 y Fx(\()p Fv(S)2918 2492 y FB(x)2962 2477 y Fv(\030)5 b Fw(j)p Fv(\030)g Fx(\))26 b Fv(f)11 b Fx(\()p Fv(S)3307 2492 y FB(x)3350 2477 y Fv(\030)5 b Fx(\))p Fv(g)t Fx(\()p Fv(\030)g Fx(\))26 b Fv(;)155 b Fx(\(34\))-24 2741 y(whic)m(h)34 b(is)g(equiv)-5 b(alen)m(t)34 b(to)e(\(32\).)43 b(Hence)34 b Fm(P)e Fx(is)h(rev)m(ersible.)123 2885 y(Due)h(to)h(Corollary)g(5)f (in)h([Ros,)g(Chapter)h(IV],)f(in)g(order)g(to)f(pro)m(v)m(e)i(ergo)s 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y Fq(1)2840 4930 y Fx(\()p Fw(N)2960 4945 y Ft(0)2999 4930 y Fv(;)17 b Fw(P)3112 4945 y Ft(0)3152 4930 y Fx(\),)32 b(whic)m(h)h(is)f (obtained)-24 5051 y(from)27 b(the)h(dominated)g(con)m(v)m(ergence)i (theorem)e(and)g(the)f(estimate)i Fw(j)p Fx(\()p Fw(T)2602 5066 y FB(t)2631 5051 y Fv(f)22 b Fw(\000)11 b Fv(f)g Fx(\)\()p Fv(\030)5 b Fx(\))p Fw(j)27 b(\024)i Fx(2)p Fw(k)p Fv(f)11 b Fw(k)3379 5066 y Fq(1)3452 5051 y Fx(\(1)g Fw(\000)g Fv(p)3687 5003 y FB(\030)3687 5073 y(t)3726 5051 y Fx(\(0)p Fw(j)p Fx(0\)\).)123 5195 y(Let)23 b(us)g(denote)h(b)m (y)f Fw(L)g Fx(the)g(generator)g(of)f(the)h(Mark)m(o)m(v)h(semigroup)g (\()p Fw(T)2660 5210 y FB(t)2690 5195 y Fx(\))2728 5210 y FB(t)p Fq(\025)p Ft(0)2870 5195 y Fx(and)f(b)m(y)h Fv(D)s Fx(\()p Fw(L)p Fx(\))j Fw(\032)h Fv(L)3603 5159 y Ft(2)3643 5195 y Fx(\()p Fw(N)3763 5210 y Ft(0)3802 5195 y Fv(;)17 b Fw(P)3915 5210 y Ft(0)3954 5195 y Fx(\))-24 5315 y(its)33 b(domain.)1935 5564 y(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop -24 171 a Fm(Prop)s(osition)38 b(4)49 b Fs(L)-5 b(et)51 b Fw(P)59 b Fs(satisfy)50 b Fv(\032)1368 186 y Ft(4)1464 171 y Fv(<)57 b Fw(1)p Fs(.)91 b(Then)50 b Fw(L)g Fs(is)g(nonp)-5 b(ositive)49 b(and)h(self{adjoint)f (with)i(c)-5 b(or)g(e)-24 291 y Fv(L)42 255 y Fq(1)117 291 y Fx(\()p Fw(N)237 306 y Ft(0)277 291 y Fv(;)17 b Fw(P)390 306 y Ft(0)429 291 y Fx(\))p Fs(.)45 b(F)-7 b(or)34 b(any)g Fv(f)39 b Fw(2)28 b Fv(L)1157 255 y Fq(1)1232 291 y Fx(\()p Fw(N)1352 306 y Ft(0)1391 291 y Fv(;)17 b Fw(P)1504 306 y Ft(0)1544 291 y Fx(\))p Fs(,)34 b(one)h(has)805 546 y Fx(\()p Fw(L)p Fv(f)11 b Fx(\)\()p Fv(\030)5 b Fx(\))54 b(=)1318 411 y Fo(Z)1445 520 y Fx(^)1434 546 y Fv(\030)5 b Fx(\()p Fv(dx)p Fx(\))28 b Fv(c)1734 561 y Ft(0)p FB(;x)1832 546 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fw(r)2066 561 y FB(x)2110 546 y Fv(f)11 b Fx(\()p Fv(\030)5 b Fx(\))27 b Fv(;)216 b Fs(for)34 b Fw(P)2787 561 y Ft(0)2827 546 y Fs(-a.e.)44 b Fv(\030)32 b(;)656 b Fx(\(36\))-24 802 y Fs(wher)-5 b(e)35 b Fw(r)335 817 y FB(x)378 802 y Fv(f)46 b Fs(is)35 b(de\014ne)-5 b(d)33 b(in)i(\(21\),)f(and,)g(mor) -5 b(e)g(over,)766 1057 y Fw(h)p Fv(f)5 b(;)17 b Fx(\()p Fw(\000L)p Fx(\))p Fv(f)11 b Fw(i)1222 1086 y Fq(P)1271 1095 y Fp(0)1365 1057 y Fx(=)1506 990 y(1)p 1506 1034 49 4 v 1506 1125 a(2)1609 921 y Fo(Z)1726 1057 y Fw(P)1795 1072 y Ft(0)1834 1057 y Fx(\()p Fv(d\030)5 b Fx(\))2026 921 y Fo(Z)2151 1031 y Fx(^)2141 1057 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))27 b Fv(c)2440 1072 y Ft(0)p FB(;x)2539 1057 y Fx(\()p Fv(\030)5 b Fx(\))44 b(\()o Fw(r)2827 1072 y FB(x)2871 1057 y Fv(f)11 b Fx(\()p Fv(\030)5 b Fx(\)\))3091 1009 y Ft(2)3175 1057 y Fv(:)617 b Fx(\(37\))-24 1336 y Fm(Pro)s(of.)86 b Fx(W)-8 b(e)44 b(use)g(the)f(abbreviation)h Fv(L)1544 1300 y FB(p)1627 1336 y Fx(for)f Fv(L)1853 1300 y FB(p)1939 1336 y Fx(=)i Fv(L)2126 1300 y FB(p)2166 1336 y Fx(\()p Fw(N)2286 1351 y Ft(0)2325 1336 y Fv(;)17 b Fw(P)2438 1351 y Ft(0)2478 1336 y Fx(\),)45 b Fv(p)h Fx(=)f(2)e(or)g Fw(1)p Fx(.)75 b(F)-8 b(or)42 b(an)m(y)i Fv(f)56 b Fw(2)46 b 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b(second)h(in)m(tegral)f(is)g(equal)-24 5195 y(to)474 5315 y Fw(\000)17 b Fm(P)645 5330 y FB(\030)683 5315 y Fx(\()g Fv(n)796 5330 y Fq(\003)835 5315 y Fx(\()p Fv(t)p Fx(\))28 b(=)f(1)17 b(\))22 b(+)g Fv(\025)1358 5330 y Ft(0)1397 5315 y Fx(\()p Fv(\030)5 b Fx(\))17 b Fv(t)54 b Fx(=)i Fw(\000)p Fx(1)22 b(+)g Fv(e)2050 5274 y Fq(\000)p FB(\025)2146 5283 y Fp(0)2181 5274 y Ft(\()p FB(\030)s Ft(\))p FB(t)2321 5315 y Fx(+)g Fv(\025)2476 5330 y Ft(0)2516 5315 y Fx(\()p Fv(\030)5 b Fx(\))17 b Fv(t)k Fx(+)h Fm(P)2888 5330 y FB(\030)2925 5315 y Fx(\()p Fv(n)3021 5330 y Fq(\003)3061 5315 y Fx(\()p Fv(t)p Fx(\))27 b Fw(\025)i Fx(2\))j(.)1935 5564 y(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop -24 171 a Fx(By)33 b(collecting)h(the)f(ab)s(o) m(v)m(e)g(estimates,)i(w)m(e)e(get)-1 377 y(1)p -14 421 75 4 v -14 513 a Fv(t)21 484 y Ft(2)99 444 y Fm(E)173 459 y Fq(P)222 468 y Fp(0)260 364 y Fo(\002\000)347 444 y Fw(T)401 459 y FB(t)431 444 y Fv(f)14 b Fw(\000)s Fv(f)g Fw(\000)s Fv(t)p Fx(\003)p 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Fx(\()p Fv(\030)p 1667 2664 48 4 v 4 w Fx(\))55 b(=)1966 2534 y Fo(X)1938 2750 y FB(s)p Fq(2)p Ft([0)p FB(;t)p Ft(])2154 2629 y Fx(\001)2235 2644 y FB(s)2272 2629 y Fx(\()p Fv(\030)p 2310 2664 V 5 w Fx(\))27 b Fv(;)1369 b Fx(\(46\))-24 2911 y(where)1315 3094 y(\001)1396 3109 y FB(s)1433 3094 y Fx(\()p Fv(\030)p 1471 3129 V 4 w Fx(\))55 b(=)1742 2924 y Fo(\()1823 3026 y Fv(x)98 b Fx(if)55 b Fv(\030)2131 3041 y FB(s)2223 3026 y Fx(=)g Fv(S)2414 3041 y FB(x)2458 3026 y Fv(\030)2501 3042 y FB(s)2534 3023 y Fk(\000)2617 3026 y Fv(;)1823 3170 y Fx(0)104 b(otherwise)29 b Fv(;)-24 3338 y Fx(and)37 b(the)f(sum)h(runs)g(o)m(v)m(er)h(all)e(jump)g(times)i Fv(s)e Fx(for)f(whic)m(h)j(\001)2242 3353 y FB(s)2279 3338 y Fx(\()p Fv(\030)p 2317 3373 V 4 w Fx(\))c Fw(6)p Fx(=)g(0.)54 b(Note)36 b(that)g(the)h(family)f Fw(f)p Fv(X)3734 3353 y Ft([)p FB(s;t)p Ft(])3889 3338 y Fx(:=)-24 3458 y Fv(X)57 3473 y FB(t)109 3458 y Fw(\000)23 b Fv(X)290 3473 y FB(s)354 3458 y Fx(:)28 b Fv(t)g(>)f(s)h Fw(\025)g Fx(0)p Fw(g)k Fx(of)g(random)h(v)-5 b(ariables)33 b(satis\014es)i(the)e (follo)m(wing)g(prop)s(erties)g(\(see)h([DF)m(GW)o(]\):)-24 3602 y(\(P1\))f Fv(X)281 3617 y Ft([)p FB(s;t)p Ft(])434 3602 y Fx(dep)s(ends)i(only)e(on)f Fw(f)p Fv(\030)1254 3617 y FB(u)1299 3602 y Fw(g)1349 3617 y FB(u)p Fq(2)p Ft([)p FB(s;t)p Ft(])1590 3602 y Fx(,)-24 3746 y(\(P2\))h(additivit)m (y:)45 b Fv(X)766 3761 y Ft([)p FB(s;t)p Ft(])909 3746 y Fx(+)22 b Fv(X)1088 3761 y Ft([)p FB(t;v)r Ft(])1241 3746 y Fx(=)28 b Fv(X)1426 3761 y Ft([)p FB(s;v)r Ft(])1590 3746 y Fx(for)k(an)m(y)i Fv(s;)17 b(t;)g(v)t(;)43 b Fx(0)28 b Fw(\024)g Fv(s)g(<)f(t)h(<)g(v)36 b Fx(,)-24 3890 y(\(P3\))d(time{co) m(v)-5 b(ariance:)45 b Fv(X)1029 3905 y Ft([)p FB(s;t)p Ft(])1172 3890 y Fw(\016)22 b Fv(\034)1286 3905 y FB(r)1380 3890 y Fx(=)55 b Fv(X)1592 3905 y Ft([)p FB(s)p Ft(+)p FB(r)n(;t)p Ft(+)p FB(r)r Ft(])1919 3890 y Fx(for)32 b(an)m(y)h Fv(r)m(;)17 b(s;)g(t)28 b Fw(\025)g Fx(0,)k Fv(s)c(<)f(t)33 b Fx(,)-24 4034 y(\(P4\))g(\()p Fv(X)319 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Ft(2)2774 5035 y Fo(\001)2864 5115 y Fv(;)928 b Fx(\(47\))-24 5315 y(whic)m(h)34 b(giv)m(es)g(a)e(w)m(a)m(y)i(to)e (calculate)h(the)g(di\013usion)g(matrix)g Fv(D)i Fx(from)d(the)h(rev)m (ersible)i(distribution)f Fm(P)e Fx(on)g(\004.)1935 5564 y(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop -24 171 a Fl(4.5)136 b(Pro)t(of)45 b(of)g(Theorem)g(1)-24 355 y Fx(The)h(lemma)g(b)s(elo)m(w)g(sho)m(ws)h (that)e(the)g Fs(me)-5 b(an)46 b(forwar)-5 b(d)45 b(velo)-5 b(city)45 b Fx(and)g(the)h Fs(in\014nitesimal)f(me)-5 b(an)45 b(squar)-5 b(e)-24 476 y(displac)g(ement)31 b Fx(de\014ned)j(as)f(in)g([DF)m(GW)o(])g(are)f(w)m(ell-de\014ned)j(in)e (the)g(presen)m(t)h(con)m(text.)-24 683 y Fm(Lemma)39 b(4)49 b Fs(L)-5 b(et)36 b Fw(P)43 b Fs(satisfy)35 b Fv(\032)1113 698 y Ft(12)1215 683 y Fv(<)28 b Fw(1)p Fs(.)-24 827 y Fx(\(i\))35 b Fs(L)-5 b(et)35 b Fv(')g Fs(b)-5 b(e)35 b(the)g Fu(R)735 791 y FB(d)775 827 y Fs(-value)-5 b(d)34 b(function)h(on)f Fw(N)1711 842 y Ft(0)1785 827 y Fs(de\014ne)-5 b(d)34 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b(e)36 b(\014rst)f(sho)m(w)h(that)f(the)g(\014rst)g(term)h (on)f(the)g(r.h.s.)51 b(v)-5 b(anishes)37 b(as)e Fv(t)c Fw(!)h Fx(0.)50 b(Using)35 b(the)h(same)f(notation)g(as)-24 3864 y(in)e(the)g(pro)s(of)f(of)g(Prop)s(osition)h(4)f(and)h(in)m(v)m (oking)h(\(39\),)128 4129 y Fm(E)202 4144 y Fq(P)251 4153 y Fp(0)290 4019 y Fo(\020)349 4015 y(\014)349 4074 y(\014)349 4134 y(\014)383 4129 y Fm(E)457 4144 y Fg(P)512 4156 y Fr(\030)550 4048 y Fo(\000)595 4129 y Fv(X)676 4144 y FB(t)723 4129 y Fv(\037)784 4048 y Fo(\000)829 4129 y Fv(n)887 4144 y Fq(\003)927 4129 y Fx(\()p Fv(t)p Fx(\))28 b(=)f(1)1218 4048 y Fo(\001\001)1332 4129 y Fw(\000)22 b Fv(t)17 b(')p Fx(\()p Fv(\030)5 b Fx(\))1671 4015 y Fo(\014)1671 4074 y(\014)1671 4134 y(\014)1703 4041 y Ft(2)1743 4019 y Fo(\021)1858 4129 y Fx(=)55 b Fm(E)2063 4144 y Fq(P)2112 4153 y Fp(0)2151 3989 y Fo(\022)2224 3985 y(\014)2224 4044 y(\014)2224 4104 y(\014)2224 4164 y(\014)2257 3994 y(Z)2384 4103 y Fx(^)2373 4129 y Fv(\030)5 b Fx(\()p Fv(dx)p 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Ft(2)3670 5004 y Fx(,)j Fv(u)c Fw(\025)h Fx(0,)-24 5124 y(and)33 b(\(41\),)f(w)m(e)i (deduce)222 5315 y Fv(I)265 5330 y Ft(1)304 5315 y Fx(\()p Fv(\030)5 b Fx(\))55 b(=)g Fw(\000)p Fx(1)23 b(+)f Fv(e)906 5274 y Fq(\000)p FB(t\025)1027 5283 y Fp(0)1062 5274 y Ft(\()p FB(\030)s Ft(\))1177 5315 y Fx(+)g Fv(t\025)1367 5330 y Ft(0)1406 5315 y Fx(\()p Fv(\030)5 b Fx(\))22 b(+)g Fm(P)1727 5330 y FB(\030)1764 5315 y Fx(\()p Fv(n)1860 5330 y Fq(\003)1900 5315 y Fx(\()p Fv(t)p Fx(\))28 b Fw(\025)g Fx(2\))55 b Fw(\024)h Fv(t)2454 5274 y Ft(2)2493 5315 y Fv(\025)2550 5330 y Ft(0)2590 5315 y Fx(\()p Fv(\030)5 b Fx(\))2714 5274 y Ft(2)2774 5315 y Fx(+)22 b Fv(t)2907 5274 y Ft(2)2947 5315 y Fv(\025)3004 5330 y Ft(0)3043 5315 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fm(E)3278 5325 y Ft(~)3268 5342 y Fg(P)3323 5354 y Fr(\030)3362 5315 y Fx(\()p Fv(\025)3457 5330 y Ft(0)3496 5315 y Fx(\()p Fv(\030)3577 5330 y Ft(1)3616 5315 y Fx(\)\))g Fv(:)1935 5564 y Fx(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop -24 171 a Fx(Moreo)m(v)m(er,)35 b Fv(I)475 186 y Ft(2)514 171 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fw(\024)h Fv(t)822 90 y Fo(R)915 144 y Fx(^)905 171 y Fv(\030)t Fx(\()p Fv(dy)t Fx(\))17 b Fv(c)1190 186 y Ft(0)p FB(;y)1285 171 y Fx(\()p Fv(\030)5 b Fx(\))p Fw(j)p Fv(y)t Fw(j)1517 134 y Ft(2)1554 171 y Fx(.)44 b(Hence)34 b(\(53\))e(is)h(b)s(ounded)g(b)m(y)247 444 y Fv(t)282 403 y Ft(3)355 304 y Fo(\022)428 444 y Fm(E)502 459 y Fq(P)551 468 y Fp(0)590 304 y Fo(\022)663 444 y Fv(\025)720 403 y Ft(2)720 469 y(0)759 444 y Fx(\()p Fv(\030)5 b Fx(\))900 309 y Fo(Z)1026 418 y Fx(^)1015 444 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))17 b Fv(c)1304 459 y Ft(0)p FB(;x)1402 444 y Fx(\()p Fv(\030)5 b Fx(\))p Fv(x)1581 403 y Ft(2)1620 304 y Fo(\023)1716 444 y Fx(+)22 b Fm(E)1888 459 y Fq(P)1937 468 y Fp(0)1976 304 y Fo(\022)2049 444 y Fv(\025)2106 459 y Ft(0)2145 444 y Fx(\()p Fv(\030)5 b Fx(\))p Fm(E)2353 454 y Ft(~)2343 471 y Fg(P)2398 483 y Fr(\030)2436 444 y Fx(\()p Fv(\025)2531 459 y Ft(0)2570 444 y Fx(\()p 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Fv(\030)5 b Fx(\))2524 962 y Fo(Z)2650 1071 y Fx(^)2640 1097 y Fv(\030)g Fx(\()p Fv(dy)t Fx(\))17 b Fv(c)2926 1112 y Ft(0)p FB(;y)3020 1097 y Fx(\()p Fv(\030)5 b Fx(\))17 b Fv(y)3213 1056 y Ft(2)3278 1097 y Fv(;)-24 1316 y Fx(and)42 b(is)f(therefore)h(b)s(ounded)g(ev)m(en)g(if)f Fv(\032)1484 1331 y Ft(4)1566 1316 y Fv(<)h Fw(1)f Fx(\(again)g(b)m(y)h(means)g(of)e (Lemma)i(1\(iv\)\).)69 b(Resuming)43 b(the)-24 1436 y(results)34 b(obtained)f(so)g(far,)f(one)h(gets)792 1623 y(1)p 779 1667 75 4 v 779 1758 a Fv(t)814 1730 y Ft(2)897 1555 y Fo(Z)1013 1690 y Fw(P)1082 1705 y Ft(0)1122 1690 y Fx(\()p Fv(d\030)5 b Fx(\))1323 1605 y Fo(\014)1323 1665 y(\014)1357 1690 y Fm(E)1431 1705 y Fg(P)1486 1717 y Fr(\030)1524 1609 y Fo(\000)1569 1690 y Fv(X)1650 1705 y FB(t)1697 1690 y Fv(\037)1758 1609 y Fo(\000)1803 1690 y Fv(n)1861 1705 y Fq(\003)1901 1690 y Fx(\()p Fv(t)p Fx(\))28 b(=)f(1)2192 1609 y Fo(\001\001)2306 1690 y Fw(\000)22 b Fv(t)17 b(')p Fx(\()p Fv(\030)5 b Fx(\))2645 1605 y Fo(\014)2645 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Fx(\()p Fv(S)2720 5281 y FB(x)2774 5239 y Fx(^)2764 5266 y Fv(\030)t Fx(\))1935 5564 y(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop -24 171 a Fx(and)305 414 y Fw(P)374 429 y Ft(0)414 414 y Fx(\()17 b Fw(j)p Fv(E)569 429 y Ft(0)608 414 y Fw(j)27 b(\024)h Fv(E)840 429 y FB(c)875 414 y Fv(;)45 b Fx(\010)1017 429 y FB(c)1052 414 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fw(2)h Fv(A)17 b Fx(\))83 b(=)1677 347 y(1)p 1676 391 51 4 v 1676 482 a Fv(\032)1753 279 y Fo(Z)1809 504 y Fq(N)1897 414 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))2166 279 y Fo(Z)2220 504 y FB(C)2270 513 y Fp(1)2336 388 y Fx(^)2326 414 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))17 b Fv(\037)p Fx(\()p Fw(j)p Fv(E)2772 429 y FB(x)2815 414 y Fw(j)27 b(\024)h Fv(E)3047 429 y FB(c)3082 414 y Fx(\))17 b Fv(\037)3198 429 y FB(A)3255 414 y Fx(\(\010)3363 429 y FB(c)3398 414 y Fx(\()p Fv(S)3496 429 y FB(x)3540 414 y Fv(\030)5 b Fx(\)\))1508 682 y(=)1677 615 y(1)p 1676 659 V 1676 750 a Fv(\032)1753 546 y Fo(Z)1809 772 y Fq(N)1897 682 y Fw(P)j Fx(\()p 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Fv(\030)t Fx(\()p Fv(dx)p Fx(\))30 b(^)-51 b Fv(c)1382 1483 y Ft(0)p FB(;x)1508 1468 y Fv(x)28 b(;)216 b Fx(\()p Fv(a)23 b Fw(\001)e Fv( )2058 1483 y FB(c)2093 1468 y Fx(\()2141 1441 y(^)2131 1468 y Fv(\030)5 b Fx(\))p Fv(a)p Fx(\))55 b(:=)2520 1332 y Fo(Z)2646 1441 y Fx(^)2636 1468 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))30 b(^)-51 b Fv(c)2935 1483 y Ft(0)p FB(;x)3061 1468 y Fx(\()p Fv(a)23 b Fw(\001)e Fv(x)p Fx(\))3315 1426 y Ft(2)3383 1468 y Fv(;)409 b Fx(\(59\))-24 1710 y Fs(as)44 b(functions)f(on)723 1685 y Fx(^)691 1710 y Fw(N)773 1725 y Ft(0)856 1710 y Fs(wher)-5 b(e)45 b Fx(^)-51 b Fv(c)1182 1725 y Ft(0)p FB(;x)1326 1710 y Fx(:=)44 b Fv(\037)p Fx(\()p Fw(j)p Fv(x)p Fw(j)g(\024)h Fv(r)1893 1725 y FB(c)1928 1710 y Fx(\))p Fs(.)72 b(Then)43 b(the)h(di\013usion)f(matrix)h Fv(D)i Fs(for)e(the)g(pr)-5 b(o)g(c)g(ess)-24 1840 y Fx(\()p Fv(X)103 1792 y FB(\030)95 1862 y(t)141 1840 y Fx(\))179 1855 y FB(t)p Fq(\025)p Ft(0)333 1840 y Fs(in)35 b(The)-5 b(or)g(em)34 b(1)g(admits)h(the)g (fol)5 b(lowing)33 b(lower)h(b)-5 b(ound)1332 2084 y Fv(D)58 b Fw(\025)1613 2016 y Fv(\032)1663 2031 y FB(c)p 1613 2061 86 4 v 1631 2152 a Fv(\032)1725 2084 y(e)1770 2043 y Fq(\000)p FB(r)1857 2051 y Fr(c)1889 2043 y Fq(\000)p Ft(4)12 b FB(\014)j(E)2098 2051 y Fr(c)2161 2084 y Fv(D)2242 2099 y FB(c)2277 2084 y Fx(\()p Fv(r)2359 2099 y FB(c)2394 2084 y Fv(;)i(E)2510 2099 y FB(c)2544 2084 y Fx(\))28 b Fv(;)-24 2349 y Fs(wher)-5 b(e)436 2495 y Fx(\()p Fv(a)23 b Fw(\001)e Fv(D)678 2510 y FB(c)713 2495 y Fx(\()p Fv(r)795 2510 y FB(c)830 2495 y Fv(;)c(E)946 2510 y FB(c)980 2495 y Fx(\))g Fv(a)p Fx(\))55 b(:=)h Fm(E)1428 2507 y Ft(^)1412 2524 y Fq(P)1467 2501 y Fr(c)1461 2545 y Fp(0)1502 2384 y Fo(\020)1578 2495 y Fx(\()p Fv(a)23 b Fw(\001)f Fv( )1803 2510 y FB(c)1838 2495 y Fv(a)p Fx(\))1944 2384 y Fo(\021)2053 2495 y Fw(\000)50 b Fx(2)2262 2359 y Fo(Z)2362 2386 y Fq(1)2317 2585 y Ft(0)2453 2495 y Fv(dt)2583 2414 y Fo(\012)2630 2495 y Fv(')2694 2510 y FB(c)2751 2495 y Fw(\001)22 b Fv(a)17 b(;)33 b(e)2974 2454 y FB(t)p Fq(L)3048 2462 y Fr(c)3101 2495 y Fv(')3165 2510 y FB(c)3222 2495 y Fw(\001)21 b Fv(a)3322 2414 y Fo(\013)3385 2517 y Ft(^)3370 2534 y Fq(P)3425 2511 y Fr(c)3419 2555 y Fp(0)3505 2495 y Fv(;)287 b Fx(\(60\))-24 2741 y Fs(and)35 b Fw(L)235 2756 y FB(c)304 2741 y Fs(is)g(the)f(unique)h(self{adjoint)f(op)-5 b(er)g(ator)34 b(on)h Fv(L)1994 2705 y Ft(2)2034 2741 y Fx(\()2104 2716 y(^)2072 2741 y Fw(N)2154 2756 y Ft(0)2193 2741 y Fv(;)2259 2716 y Fx(^)2237 2741 y Fw(P)2314 2705 y FB(c)2306 2766 y Ft(0)2349 2741 y Fx(\))g Fs(such)f(that)763 3010 y Fx(\()p Fw(L)870 3025 y FB(c)904 3010 y Fv(f)11 b Fx(\)\()1049 2984 y(^)1039 3010 y Fv(\030)t Fx(\))56 b(=)1311 2874 y Fo(Z)1437 2984 y Fx(^)1427 3010 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))30 b(^)-51 b Fv(c)1726 3025 y Ft(0)p FB(;x)1825 3010 y Fw(r)1908 3025 y FB(x)1952 3010 y Fv(f)11 b Fx(\()2059 2984 y(^)2049 3010 y Fv(\030)t Fx(\))27 b Fv(;)216 b Fw(8)28 b Fv(f)39 b Fw(2)28 b Fv(L)2734 2969 y Fq(1)2809 3010 y Fx(\()2879 2985 y(^)2847 3010 y Fw(N)2929 3025 y Ft(0)2968 3010 y Fv(;)3034 2985 y Fx(^)3012 3010 y Fw(P)3089 2969 y FB(c)3081 3035 y Ft(0)3124 3010 y Fx(\))17 b Fv(:)613 b Fx(\(61\))123 3273 y(One)39 b(can)g(pro)m(v)m(e)h(b)m(y)g(the)f(same)h(argumen)m(ts)f(used)h(in)f (the)h(pro)s(of)e(of)g(Prop)s(osition)h(4)f(that)h(the)g(ab)s(o)m(v)m (e)-24 3393 y(op)s(erator)d Fw(L)442 3408 y FB(c)513 3393 y Fx(is)i(w)m(ell-de\014ned)h(and)d(self{adjoin)m(t.)57 b(W)-8 b(e)37 b(will)g(write)2543 3368 y(^)2529 3393 y Fm(P)2606 3357 y FB(c)2677 3393 y Fx(for)f(the)h(probabilit)m(y)h (measure)g(on)-24 3514 y(the)h(path)g(space)659 3488 y(^)651 3514 y(\004)f(:=)g Fv(D)s Fx(\()17 b([0)p Fv(;)g Fw(1)p Fx(\))p Fv(;)1366 3488 y Fx(^)1336 3514 y Fw(N)1418 3529 y Ft(0)1472 3514 y Fx(\))38 b(asso)s(ciated)i(to)e(the)h(Mark)m(o) m(v)h(pro)s(cess)g(with)f(generator)g Fw(L)3762 3529 y FB(c)3835 3514 y Fx(and)-24 3634 y(initial)31 b(distribution)809 3609 y(^)786 3634 y Fw(P)863 3598 y FB(c)855 3659 y Ft(0)898 3634 y Fx(.)43 b(If)31 b(the)g(initial)f(distribution)i(is)f Fv(\016)2185 3646 y Ft(^)2178 3663 y FB(\030)2216 3634 y Fx(,)g(with)2505 3608 y(^)2495 3634 y Fv(\030)h Fw(2)2696 3609 y Fx(^)2664 3634 y Fw(N)2746 3649 y Ft(0)2785 3634 y Fx(,)f(then)g(w)m(e)h(will)f(write)3648 3609 y(^)3635 3634 y Fm(P)3712 3598 y FB(c)3719 3660 y Ft(^)3712 3677 y FB(\030)3749 3634 y Fx(.)43 b(One)-24 3793 y(can)e(pro)m(v)m(e)g (that)f(these)i(Mark)m(o)m(v)f(pro)s(cesses)h(are)f(w)m(ell)g (de\014ned)h(\(in)e(particular,)3080 3768 y(^)3066 3793 y Fm(P)3143 3757 y FB(c)3150 3819 y Ft(^)3143 3837 y FB(\030)3221 3793 y Fx(is)h(w)m(ell)g(de\014ned)h(for)-1 3928 y(^)-24 3953 y Fw(P)53 3917 y FB(c)45 3978 y Ft(0)89 3953 y Fx({almost)33 b(all)600 3927 y(^)590 3953 y Fv(\030)t Fx(\))h(and)g(exhibit)h(a)e(realization)i(as)e(jump)i(pro)s(cesses)g(b) m(y)g(means)g(of)e(the)h(same)h(argumen)m(ts)-24 4074 y(used)41 b(in)e(Section)h(3)f(\(note)g(that,)h(for)f(a)g(suitable)h(p) s(ositiv)m(e)g(constan)m(t)g Fv(c)p Fx(,)2760 3994 y Fo(R)2853 4048 y Fx(^)2843 4074 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))r(^)-51 b Fv(c)3114 4089 y Ft(0)p FB(;x)3251 4074 y Fw(\024)40 b Fv(c)17 b(\025)3484 4089 y Ft(0)3523 4074 y Fx(\()p Fv(\030)5 b Fx(\))38 b(for)g(an)m(y)-24 4194 y Fv(\030)32 b Fw(2)d(N)228 4209 y Ft(0)267 4194 y Fx(,)g(th)m(us)g (allo)m(wing)g(to)f(exclude)i(explosion)g(phenomena)f(from)f(the)h (results)g(of)f(Section)h(3.1\).)42 b(Finally)-8 b(,)-24 4315 y(giv)m(en)241 4288 y(^)231 4315 y Fv(\030)p 231 4350 48 4 v 32 w Fw(2)408 4289 y Fx(^)400 4315 y(\004,)33 b Fv(X)606 4330 y FB(t)636 4315 y Fx(\()684 4288 y(^)674 4315 y Fv(\030)p 674 4350 V 4 w Fx(\))f(is)h(de\014ned)h(as)f(in)g (\(46\).)-24 4482 y Fm(Pro)s(of.)50 b Fx(Note)33 b(that)1381 4603 y Fv(c)1423 4618 y Ft(0)p FB(;x)1521 4603 y Fx(\()p Fv(\030)5 b Fx(\))55 b Fw(\025)h Fv(e)1878 4561 y Fq(\000)p FB(r)1965 4569 y Fr(c)1997 4561 y Fq(\000)p Ft(4)12 b FB(\014)j(E)2206 4569 y Fr(c)2271 4603 y Fx(~)-51 b Fv(c)2311 4618 y Ft(0)p FB(;x)2410 4603 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fv(;)-24 4777 y Fx(where)918 4897 y(~)-51 b Fv(c)958 4912 y FB(x;y)1058 4897 y Fx(\()p Fv(\030)5 b Fx(\))55 b(:=)g Fv(\037)1456 4817 y Fo(\000)1502 4897 y Fw(j)p Fv(E)1602 4912 y FB(x)1646 4897 y Fw(j)27 b(\024)h Fv(E)1878 4912 y FB(c)1913 4897 y Fv(;)45 b Fw(j)p Fv(E)2085 4912 y FB(y)2126 4897 y Fw(j)27 b(\024)i Fv(E)2359 4912 y FB(c)2393 4897 y Fv(;)45 b Fw(j)p Fv(x)22 b Fw(\000)h Fv(y)t Fw(j)j(\024)i Fv(r)2925 4912 y FB(c)2960 4897 y Fx(\))g Fv(:)-24 5072 y Fx(Then)34 b(\(20\))e(implies)1416 5192 y(\()p Fv(a)23 b Fw(\001)f Fv(D)s(a)p Fx(\))55 b Fw(\025)h Fv(e)1984 5151 y Fq(\000)p FB(r)2071 5159 y Fr(c)2102 5151 y Fq(\000)p Ft(4)12 b FB(\014)j(E)2311 5159 y Fr(c)2375 5192 y Fv(g)t Fx(\()p Fv(a)p Fx(\))1935 5564 y(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop -24 171 a Fx(where)466 314 y Fv(g)t Fx(\()p Fv(a)p Fx(\))55 b(:=)222 b(inf)857 381 y FB(f)7 b Fq(2)p FB(L)993 362 y Fk(1)1058 381 y Ft(\()p Fq(N)1143 390 y Fp(0)1178 381 y FB(;)p Fq(P)1247 390 y Fp(0)1281 381 y Ft(\))1326 179 y Fo(Z)1442 314 y Fw(P)1511 329 y Ft(0)1550 314 y Fx(\()p Fv(d\030)e Fx(\))1758 179 y Fo(Z)1884 288 y Fx(^)1874 314 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))30 b(~)-51 b Fv(c)2173 329 y Ft(0)p FB(;x)2272 314 y Fx(\()p Fv(\030)5 b Fx(\))2423 204 y Fo(\020)2482 314 y Fv(a)23 b Fw(\001)e Fv(x)40 b Fx(+)e Fw(r)2897 329 y FB(x)2941 314 y Fv(f)11 b Fx(\()p Fv(\030)5 b Fx(\))3141 204 y Fo(\021)3199 226 y Ft(2)3294 314 y Fw(\025)28 b Fx(0)g Fv(:)-24 519 y Fx(By)k(the)g(same)g(argumen)m(ts)g(used)g(in)g (the)f(pro)s(of)g(of)g(Prop)s(osition)g(4)g(one)g(can)h(sho)m(w)g(that) f(there)h(is)g(a)f(unique)-24 640 y(self{adjoin)m(t)i(op)s(erator)911 614 y(~)887 640 y Fw(L)g Fx(on)f Fv(L)1190 603 y Ft(2)1230 640 y Fx(\()p Fw(N)1350 655 y Ft(0)1389 640 y Fv(;)17 b Fw(P)1502 655 y Ft(0)1541 640 y Fx(\))33 b(suc)m(h)h(that)703 862 y(\()765 837 y(~)741 862 y Fw(L)p Fv(f)11 b Fx(\)\()p Fv(\030)5 b Fx(\))54 b(:=)1244 726 y Fo(Z)1370 835 y Fx(^)1360 862 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))30 b(~)-51 b Fv(c)1659 877 y Ft(0)p FB(;x)1758 862 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fw(r)1992 877 y FB(x)2036 862 y Fv(f)11 b Fx(\()p Fv(\030)5 b Fx(\))26 b Fv(;)212 b Fw(8)28 b Fv(f)38 b Fw(2)29 b Fv(L)2814 821 y Fq(1)2889 862 y Fx(\()p Fw(N)3009 877 y Ft(0)3048 862 y Fv(;)17 b Fw(P)3161 877 y Ft(0)3200 862 y Fx(\))p Fv(:)-24 1101 y Fx(Moreo)m(v)m(er,)35 b Fv(L)498 1064 y Fq(1)573 1101 y Fx(\()p Fw(N)693 1116 y Ft(0)732 1101 y Fv(;)17 b Fw(P)845 1116 y Ft(0)884 1101 y Fx(\))33 b(is)g(a)f(core)h(of)1475 1075 y(~)1451 1101 y Fw(L)f Fx(and)169 1323 y Fw(h)p Fv(f)5 b(;)17 b Fx(\()p Fw(\000)444 1298 y Fx(~)420 1323 y Fw(L)p Fx(\))p Fv(f)11 b Fw(i)625 1352 y Fq(P)674 1361 y Fp(0)768 1323 y Fx(=)909 1255 y(1)p 909 1300 49 4 v 909 1391 a(2)1012 1187 y Fo(Z)1129 1323 y Fw(P)1198 1338 y Ft(0)1237 1323 y Fx(\()p Fv(d\030)5 b Fx(\))1429 1187 y Fo(Z)1554 1296 y Fx(^)1544 1323 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))29 b(~)-51 b Fv(c)1843 1338 y Ft(0)p FB(;x)1942 1323 y Fx(\()p Fv(\030)5 b Fx(\))44 b(\()o Fw(r)2230 1338 y FB(x)2274 1323 y Fv(f)11 b Fx(\()p Fv(\030)5 b Fx(\)\))2494 1274 y Ft(2)2578 1323 y Fv(;)212 b Fw(8)28 b Fv(f)38 b Fw(2)28 b Fv(L)3146 1282 y Fq(1)3221 1323 y Fx(\()p Fw(N)3341 1338 y Ft(0)3381 1323 y Fv(;)17 b Fw(P)3494 1338 y Ft(0)3533 1323 y Fx(\))28 b Fv(:)193 b Fx(\(62\))123 1539 y(Next)33 b(let)g(us)g(in)m(tro)s(duce)h(the)f(functions)499 1737 y(~)-64 b Fv(')o Fx(\()p Fv(\030)5 b Fx(\))55 b(=)857 1601 y Fo(Z)984 1710 y Fx(^)973 1737 y Fv(\030)5 b Fx(\()p Fv(dx)p Fx(\))30 b(~)-51 b Fv(c)1273 1752 y Ft(0)p FB(;x)1371 1737 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fv(x)h(;)212 b Fx(\()p Fv(a)22 b Fw(\001)2025 1710 y Fx(~)2005 1737 y Fv( )t Fx(\()p Fv(\030)5 b Fx(\))p Fv(a)p Fx(\))55 b(=)2471 1601 y Fo(Z)2598 1710 y Fx(^)2588 1737 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))30 b(~)-51 b Fv(c)2887 1752 y Ft(0)p FB(;x)2985 1737 y Fx(\()p Fv(\030)5 b Fx(\))27 b(\()p Fv(a)c Fw(\001)e Fv(x)p Fx(\))3390 1696 y Ft(2)3458 1737 y Fv(:)-24 1959 y Fx(Then,)41 b(w)m(e)e(obtain)f(b)m(y)h(means)g(of)f (straigh)m(tforw)m(ard)g(computations)h(and)f(the)h(iden)m(tities)h (\(44\),)f(\(45\))e(and)-24 2079 y(\(62\))32 b(that)493 2281 y Fv(g)t Fx(\()p Fv(a)p Fx(\))54 b(=)i Fm(E)931 2296 y Fq(P)980 2305 y Fp(0)1018 2170 y Fo(\020)1094 2281 y Fx(\()p Fv(a)23 b Fw(\001)1275 2254 y Fx(~)1255 2281 y Fv( )t(a)p Fx(\))1428 2170 y Fo(\021)1510 2281 y Fw(\000)f Fx(2)197 b(sup)1703 2366 y FB(f)7 b Fq(2)p FB(L)1839 2347 y Fk(1)1904 2366 y Ft(\()p Fq(N)1989 2375 y Fp(0)2024 2366 y FB(;)p Fq(P)2093 2375 y Fp(0)2127 2366 y Ft(\))2155 2170 y Fo(\020)2214 2281 y Fx(2)33 b Fw(h)16 b Fx(~)-65 b Fv(')22 b Fw(\001)g Fv(a;)17 b(f)11 b Fw(i)2663 2310 y Fq(P)2712 2319 y Fp(0)2773 2281 y Fw(\000)23 b(h)p Fv(f)5 b(;)17 b Fx(\()p Fw(\000)3148 2255 y Fx(~)3124 2281 y Fw(L)p Fx(\))p Fv(f)11 b Fw(i)3329 2296 y Fq(P)3378 2305 y Fp(0)3416 2170 y Fo(\021)725 2564 y Fx(=)56 b Fm(E)931 2579 y Fq(P)980 2588 y Fp(0)1018 2453 y Fo(\020)1105 2564 y Fx(\()p Fv(a)23 b Fw(\001)1286 2538 y Fx(~)1267 2564 y Fv( )t(a)p Fx(\))1450 2453 y Fo(\021)1560 2564 y Fw(\000)50 b Fx(2)1769 2428 y Fo(Z)1869 2455 y Fq(1)1824 2654 y Ft(0)1960 2564 y Fv(dt)2090 2453 y Fo(D)2167 2564 y Fx(~)-65 b Fv(')22 b Fw(\001)g Fv(a)17 b(;)33 b(e)2460 2523 y FB(t)2502 2506 y Ft(~)2485 2523 y Fq(L)2571 2564 y Fx(~)-65 b Fv(')22 b Fw(\001)g Fv(a)2742 2453 y Fo(E)2803 2633 y Fq(P)2852 2642 y Fp(0)2908 2564 y Fv(:)-24 2790 y Fx(A)m(t)33 b(this)g(p)s(oin)m(t,)g(in)g(order)g(to)f (get)h(\(60\),)f(it)g(is)h(enough)g(to)g(sho)m(w)g(that)1168 2992 y Fm(E)1242 3007 y Fq(P)1291 3016 y Fp(0)1330 2881 y Fo(\020)1417 2992 y Fx(\()p Fv(a)23 b Fw(\001)1598 2965 y Fx(~)1578 2992 y Fv( )t(a)p Fx(\))1762 2881 y Fo(\021)1849 2992 y Fx(=)1963 2924 y Fv(\032)2013 2939 y FB(c)p 1963 2969 86 4 v 1980 3060 a Fv(\032)2058 2992 y Fm(E)2148 3004 y Ft(^)2132 3020 y Fq(P)2187 2997 y Fr(c)2181 3041 y Fp(0)2223 2881 y Fo(\020)2310 2992 y Fx(\()p Fv(a)f Fw(\001)g Fv( )2534 3007 y FB(c)2569 2992 y Fv(a)p Fx(\))2686 2881 y Fo(\021)2773 2992 y Fv(;)-24 3210 y Fx(and)983 3220 y Fo(D)1060 3331 y Fx(~)-65 b Fv(')22 b Fw(\001)g Fv(a)17 b(;)33 b(e)1353 3289 y FB(t)1395 3273 y Ft(~)1378 3289 y Fq(L)1463 3331 y Fx(~)-64 b Fv(')22 b Fw(\001)g Fv(a)1635 3220 y Fo(E)1696 3400 y Fq(P)1745 3409 y Fp(0)1812 3331 y Fx(=)1925 3263 y Fv(\032)1975 3278 y FB(c)p 1925 3308 V 1942 3399 a Fv(\032)2037 3250 y Fo(\012)2084 3331 y Fv(')2148 3346 y FB(c)2204 3331 y Fw(\001)g Fv(a)17 b(;)33 b(e)2427 3289 y FB(t)p Fq(L)2501 3297 y Fr(c)2554 3331 y Fv(')2618 3346 y FB(c)2675 3331 y Fw(\001)22 b Fv(a)2776 3250 y Fo(\013)2839 3353 y Ft(^)2823 3370 y Fq(P)2878 3347 y Fr(c)2872 3391 y Fp(0)2958 3331 y Fv(:)-24 3530 y Fx(This)34 b(can)f(b)s(e)g(deriv)m(ed)h(from)e(Lemma) i(5)e(and)h(the)g(iden)m(tities)1491 3798 y(~)1471 3824 y Fv( )87 b Fx(=)c Fv(\037)p Fx(\()p Fw(j)p Fv(E)1979 3839 y Ft(0)2019 3824 y Fw(j)27 b(\024)h Fv(E)2251 3839 y FB(c)2286 3824 y Fx(\))17 b Fv( )2404 3839 y FB(c)2461 3824 y Fw(\016)22 b Fx(\010)2603 3839 y FB(c)2666 3824 y Fv(;)1490 3969 y Fx(~)-65 b Fv(')83 b Fx(=)g Fv(\037)p Fx(\()p Fw(j)p Fv(E)1979 3984 y Ft(0)2019 3969 y Fw(j)27 b(\024)h Fv(E)2251 3984 y FB(c)2286 3969 y Fx(\))17 b Fv(')2405 3984 y FB(c)2461 3969 y Fw(\016)22 b Fx(\010)2603 3984 y FB(c)2666 3969 y Fv(;)1160 4090 y Fx(~)1136 4115 y Fw(L)p Fx(\()p Fv(f)32 b Fw(\016)22 b Fx(\010)1465 4130 y FB(c)1500 4115 y Fx(\))83 b(=)g Fv(\037)p Fx(\()p Fw(j)p Fv(E)1979 4130 y Ft(0)2019 4115 y Fw(j)27 b(\024)h Fv(E)2251 4130 y FB(c)2286 4115 y Fx(\))17 b(\()p Fw(L)2448 4130 y FB(c)2482 4115 y Fv(f)11 b Fx(\))22 b Fw(\016)g Fx(\010)2743 4130 y FB(c)2806 4115 y Fv(;)-24 4289 y Fx(where)34 b(\010)328 4304 y FB(c)396 4289 y Fx(has)f(b)s(een)g(in)m (tro)s(duced)h(in)f(\(56\).)2315 b Ff(2)-24 4614 y Fy(6)161 b(P)l(erio)t(dic)53 b(appro)l(ximan)l(ts)f(and)i(resistor)e(net)l(w)l (orks)-24 4834 y Fx(In)29 b(this)g(section,)h(w)m(e)g(compare)f Fv(D)1236 4849 y FB(c)1271 4834 y Fx(\()p Fv(r)1353 4849 y FB(c)1387 4834 y Fv(;)17 b(E)1503 4849 y FB(c)1538 4834 y Fx(\))28 b(to)g(the)h(di\013usion)g(co)s(e\016cien)m(t)h(of)e (adequately)i(de\014ned)g(p)s(erio)s(dic)-24 4954 y(appro)m(ximan)m (ts,)41 b(whic)m(h)e(then)f(in)f(turn)h(can)f(b)s(e)h(calculated)g(as)g (the)g(conductance)h(of)e(a)g(random)g(resistor)-24 5074 y(net)m(w)m(ork.)48 b(There)35 b(ha)m(v)m(e)f(b)s(een)g(n)m(umerous)i (w)m(orks)e(on)g(p)s(erio)s(dic)f(appro)m(ximan)m(ts;)j(a)d(recen)m(t)i (one)e(con)m(taining)-24 5195 y(further)e(references)g(is)g([Owh].)43 b(Still,)31 b(w)m(e)g(merely)g(supp)s(ose)g(that)f(the)g Fs(r)-5 b(andom)31 b(envir)-5 b(onment)31 b(satis\014es)h(the)-24 5315 y(assumptions)i(of)h(The)-5 b(or)g(em)34 b(1)g(as)h(wel)5 b(l)34 b(as)h(the)g(isotr)-5 b(opy)34 b(pr)-5 b(op)g(erty)35 b(of)g(R)-5 b(emark)34 b(4)p Fx(.)1935 5564 y(27)p eop end %%Page: 28 28 TeXDict begin 28 27 bop -24 171 a Fl(6.1)136 b(Random)45 b(w)l(alk)h(on)f(a)g(p)t(erio)t(dized)g(medium)-24 355 y Fx(Let)23 b(us)h(c)m(ho)s(ose)g(a)e(giv)m(en)i(direction)g(in)f Fu(R)1446 319 y FB(d)1487 355 y Fx(,)h(sa)m(y)-8 b(,)26 b(the)d(direction)h(parallel)f(to)g(the)g(axis)h(of)e(the)i(\014rst)f (co)s(ordinate.)-24 476 y(Giv)m(en)34 b(a)e(\014xed)i(con\014guration) 1177 449 y(^)1167 476 y Fv(\030)e Fw(2)1368 451 y Fx(^)1336 476 y Fw(N)47 b Fx(and)32 b Fv(N)39 b(>)27 b(r)1918 491 y FB(c)1953 476 y Fx(,)32 b(w)m(e)i(de\014ne)g(the)f(follo)m(wing)g (subsets)h(of)e Fu(R)3538 440 y FB(d)142 714 y Fv(Q)226 649 y Ft(^)219 667 y FB(\030)219 741 y(N)342 714 y Fx(:=)55 b(supp)q(\()749 688 y(^)739 714 y Fv(\030)5 b Fx(\))22 b Fw(\\)g Fv(C)1005 729 y Ft(2)p FB(N)1135 714 y Fv(;)212 b Fx(\000)1435 673 y Fq(\006)1435 741 y FB(N)1558 714 y Fx(:=)55 b Fu(Z)1782 673 y FB(d)1845 714 y Fw(\\)23 b(f)p Fv(x)k Fx(:)h Fv(x)2176 673 y Ft(\(1\))2299 714 y Fx(=)f Fw(\006)p Fv(N)5 b(;)45 b Fw(j)p Fv(x)2717 673 y Ft(\()p FB(j)t Ft(\))2809 714 y Fw(j)27 b Fv(<)h(N)43 b Fx(for)32 b Fv(j)h Fx(=)28 b(2)p Fv(;)17 b(:)g(:)g(:)e(;)i(d)p Fw(g)g Fv(;)p 142 829 70 4 v 142 909 a Fw(V)218 829 y Ft(^)211 847 y FB(\030)211 933 y(N)334 909 y Fx(:=)55 b Fv(Q)576 844 y Ft(^)569 861 y FB(\030)569 936 y(N)659 909 y Fw([)22 b Fx(\000)808 867 y Ft(+)808 936 y FB(N)898 909 y Fw([)g Fx(\000)1047 867 y Fq(\000)1047 936 y FB(N)1142 909 y Fv(;)212 b(B)1467 844 y Ft(^)1460 861 y FB(\030)s Fq(\006)1455 936 y FB(N)1608 909 y Fx(:=)55 b Fv(Q)1850 844 y Ft(^)1843 861 y FB(\030)1843 936 y(N)1933 909 y Fw(\\)23 b Fv(B)2101 867 y Fq(\006)2096 936 y FB(N)2191 909 y Fv(;)-24 1139 y Fx(where)42 b(as)f(b)s(efore)f Fv(C)764 1154 y Ft(2)p FB(N)907 1139 y Fx(=)h(\()p Fw(\000)p Fv(N)5 b(;)17 b(N)10 b Fx(\))1392 1102 y FB(d)1474 1139 y Fx(as)40 b(w)m(ell)i(as)e Fv(B)2013 1097 y Fq(\000)2008 1166 y FB(N)2117 1139 y Fx(=)h Fw(f)p Fv(x)g Fw(2)g Fv(C)2557 1154 y Ft(2)p FB(N)2717 1139 y Fx(:)58 b Fv(x)2857 1102 y Ft(\(1\))2993 1139 y Fw(2)41 b Fx(\()p Fw(\000)p Fv(N)5 b(;)17 b Fw(\000)p Fv(N)39 b Fx(+)27 b Fv(r)3683 1154 y FB(c)3718 1139 y Fx(])p Fw(g)40 b Fx(and)-24 1259 y Fv(B)55 1218 y Ft(+)50 1286 y FB(N)145 1259 y Fx(=)28 b Fw(f)p Fv(x)g Fw(2)g Fv(C)546 1274 y Ft(2)p FB(N)693 1259 y Fx(:)44 b Fv(x)819 1223 y Ft(\(1\))941 1259 y Fw(2)28 b Fx([)p Fv(N)33 b Fw(\000)23 b Fv(r)1317 1274 y FB(c)1352 1259 y Fv(;)17 b(N)10 b Fx(\))p Fw(g)p Fx(.)123 1467 y(Next)36 b(let)f(us)h(in)m(tro)s(duce)g(a)f(graph)g(\()p 1472 1387 V Fw(V)1548 1388 y Ft(^)1541 1405 y FB(\030)1541 1491 y(N)1608 1467 y Fv(;)p 1652 1387 62 4 v 17 w Fw(E)1721 1388 y Ft(^)1714 1405 y FB(\030)1714 1491 y(N)1781 1467 y Fx(\))g(with)h(set)f(of)g(v)m(ertices)p 2703 1387 70 4 v 38 w Fw(V)2779 1388 y Ft(^)2772 1405 y FB(\030)2772 1491 y(N)2874 1467 y Fx(and)g(set)h(of)f(edges)p 3598 1387 62 4 v 36 w Fw(E)3667 1388 y Ft(^)3660 1405 y FB(\030)3660 1491 y(N)3727 1467 y Fx(.)51 b(Tw)m(o)-24 1625 y(v)m(ertices)32 b Fv(x;)17 b(y)31 b Fw(2)d Fv(Q)682 1561 y Ft(^)675 1578 y FB(\030)675 1652 y(N)773 1625 y Fx(are)i(connected)h(b)m(y)g(a)e (non-orien)m(ted)i(edge)f Fw(f)p Fv(x;)17 b(y)t Fw(g)27 b(2)p 2751 1545 V 28 w(E)2819 1546 y Ft(^)2812 1564 y FB(\030)2812 1650 y(N)2909 1625 y Fx(if)j(and)g(only)g(if)g Fw(j)p Fv(x)17 b Fw(\000)g Fv(y)t Fw(j)26 b(\024)i Fv(r)3930 1640 y FB(c)3965 1625 y Fx(;)-24 1772 y(moreo)m(v)m(er,)k(all)d(v)m (ertices)j Fv(x)c Fw(2)g Fv(B)1166 1707 y Ft(^)1159 1724 y FB(\030)s Ft(+)1154 1799 y FB(N)1282 1772 y Fx(\(resp)s(ectiv)m(ely)k Fv(x)c Fw(2)g Fv(B)2111 1707 y Ft(^)2104 1724 y FB(\030)s Fq(\000)2099 1799 y FB(N)2197 1772 y Fx(\))h(are)h(connected)h(to)e (all)h Fv(y)g Fw(2)e Fx(\000)3358 1730 y Ft(+)3358 1799 y FB(N)3455 1772 y Fx(\(resp)s(ectiv)m(ely)-24 1932 y Fv(y)j Fw(2)d Fx(\000)210 1891 y Fq(\000)210 1959 y FB(N)277 1932 y Fx(\))33 b(b)m(y)g(an)g(edge)g Fw(f)p Fv(x;)17 b(y)t Fw(g)27 b(2)p 1213 1852 V 28 w(E)1281 1853 y Ft(^)1274 1871 y FB(\030)1274 1957 y(N)1374 1932 y Fx(and)33 b(the)g(p)s(oin)m (ts)g(of)f(\000)2197 1891 y Fq(\006)2197 1959 y FB(N)2297 1932 y Fx(are)g(not)h(connected)h(b)s(et)m(w)m(een)h(themselv)m(es.)123 2140 y(W)-8 b(e)33 b(no)m(w)g(de\014ne)h(another)e(graph)h(\()p Fw(V)1524 2075 y Ft(^)1517 2093 y FB(\030)1509 2167 y(N)1576 2140 y Fv(;)17 b Fw(E)1689 2075 y Ft(^)1682 2093 y FB(\030)1673 2167 y(N)1740 2140 y Fx(\))32 b(obtained)h(from)f(\()p 2479 2060 70 4 v Fw(V)2556 2061 y Ft(^)2549 2079 y FB(\030)2549 2165 y(N)2616 2140 y Fv(;)p 2660 2060 62 4 v 17 w Fw(E)2728 2061 y Ft(^)2721 2079 y FB(\030)2721 2165 y(N)2788 2140 y Fx(\))h(b)m(y)g(iden)m(tifying)h(the)f(v)m(ertices)622 2360 y Fv(x)677 2375 y Fq(\000)791 2360 y Fx(=)56 b(\()p Fw(\000)p Fv(N)5 b(;)17 b(x)1220 2319 y Ft(\(2\))1315 2360 y Fv(;)g(:)g(:)g(:)f(;)h(x)1589 2319 y Ft(\()p FB(d)p Ft(\))1684 2360 y Fx(\))195 b(and)g Fv(x)2324 2375 y Ft(+)2439 2360 y Fx(=)55 b(\()p Fv(N)5 b(;)17 b(x)2790 2319 y Ft(\(2\))2885 2360 y Fv(;)g(:)g(:)g(:)f(;)h(x)3159 2319 y Ft(\()p FB(d)p Ft(\))3254 2360 y Fx(\))28 b Fv(:)-24 2618 y Fx(Let)36 b(us)h(write)g Fv(\031)g Fx(:)p 705 2538 70 4 v 50 w Fw(V)781 2539 y Ft(^)774 2557 y FB(\030)774 2643 y(N)875 2618 y Fw(!)c(V)1084 2554 y Ft(^)1077 2571 y FB(\030)1069 2645 y(N)1172 2618 y Fx(for)i(the)i(iden)m(ti\014cation) g(map)f(on)g(the)g(sets)h(of)f(v)m(ertices.)56 b(Hence)37 b Fv(\031)t Fx(\(\000)3778 2577 y Fq(\000)3778 2645 y FB(N)3845 2618 y Fx(\))c(=)-24 2779 y Fv(\031)t Fx(\(\000)134 2738 y Ft(+)134 2806 y FB(N)201 2779 y Fx(\))c(and)g Fv(\031)j Fx(restricted)e(to)e Fv(Q)1170 2714 y Ft(^)1163 2732 y FB(\030)1163 2806 y(N)1259 2779 y Fx(is)i(the)f(iden)m(tit)m(y)h (map.)42 b(The)30 b(set)f Fw(V)2548 2714 y Ft(^)2541 2732 y FB(\030)2533 2806 y(N)2628 2779 y Fx(=)f Fv(\031)t Fx(\()p 2829 2699 V Fw(V)2905 2700 y Ft(^)2898 2717 y FB(\030)2898 2804 y(N)2965 2779 y Fx(\))h(represen)m(ts)i(the)e(medium) -24 2899 y(p)s(erio)s(dized)34 b(along)e(the)h(\014rst)g(co)s (ordinate.)123 3085 y(No)m(w)d(a)f(con)m(tin)m(uous{time)j(random)d(w)m (alk)i(with)f(state)g(space)h Fw(V)2512 3021 y Ft(^)2505 3038 y FB(\030)2497 3112 y(N)2594 3085 y Fx(and)e(in\014nitesimal)j (generator)d Fw(L)3837 3021 y Ft(^)3830 3038 y FB(\030)3830 3112 y(N)3927 3085 y Fx(is)-24 3206 y(giv)m(en)34 b(b)m(y)555 3264 y Fo(\000)601 3344 y Fw(L)677 3279 y Ft(^)670 3297 y FB(\030)670 3371 y(N)737 3344 y Fv(f)796 3264 y Fo(\001)841 3344 y Fx(\()p Fv(x)p Fx(\))56 b(=)1356 3250 y Fo(X)1159 3496 y FB(y)r Fq(2V)1299 3450 y Fp(^)1293 3463 y Fr(\030)1287 3519 y(N)1357 3496 y Ft(:)11 b Fq(f)p FB(x;y)r Fq(g2E)1651 3450 y Fp(^)1645 3463 y Fr(\030)1639 3519 y(N)1714 3344 y Fv(c)p Fx(\()p Fw(f)p Fv(x;)17 b(y)t Fw(g)p Fx(\))2083 3264 y Fo(\000)2144 3344 y Fv(f)11 b Fx(\()p Fv(y)t Fx(\))21 b Fw(\000)h Fv(f)11 b Fx(\()p Fv(x)p Fx(\))2658 3264 y Fo(\001)2731 3344 y Fv(;)212 b Fw(8)28 b Fv(x)g Fw(2)g(V)3306 3279 y Ft(^)3299 3297 y FB(\030)3291 3371 y(N)3386 3344 y Fv(;)-24 3702 y Fx(where)34 b(the)f(b)s(ond-dep)s(enden)m(t)h (transition)f(rates)g Fv(c)p Fx(\()p Fw(f)p Fv(x;)17 b(y)t Fw(g)p Fx(\))31 b(are)i(de\014ned)h(for)e(an)m(y)h Fw(f)p Fv(x;)17 b(y)t Fw(g)27 b(2)h(E)3501 3637 y Ft(^)3494 3655 y FB(\030)3485 3729 y(N)3584 3702 y Fx(b)m(y)903 4023 y Fv(c)p Fx(\()p Fw(f)p Fv(x;)17 b(y)t Fw(g)p 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b(x)28 b Fw(2)g Fv(\031)t Fx(\(\000)3074 1340 y Fq(\000)3074 1408 y FB(N)3141 1381 y Fx(\))f Fv(:)-24 1599 y Fx(Finally)-8 b(,)33 b(giv)m(en)h Fv(t)28 b Fw(\025)g Fx(0,)k(w)m(e)i(de\014ne)g(the)f(random)g(v)-5 b(ariable)1197 1831 y Fv(X)1286 1780 y Ft(\(1\))1382 1763 y(^)1375 1780 y FB(\030)1278 1858 y(N)s(;t)1414 1831 y Fx(\()p Fv(!)p 1452 1847 65 4 v 3 w Fx(\))56 b(=)1919 1736 y Fo(X)1741 1952 y FB(s)p Fq(2)p Ft([0)p FB(;t)p Ft(])11 b(:)g FB(!)2027 1960 y Fr(s)2060 1952 y Fq(6)p Ft(=)p FB(!)2159 1961 y Fr(s)p Fk(\000)2257 1831 y Fv(d)2308 1846 y Ft(1)2347 1831 y Fx(\()p Fv(!)2446 1846 y FB(s)p Fq(\000)2537 1831 y Fv(;)17 b(!)2642 1846 y FB(s)2679 1831 y Fx(\))27 b Fv(;)-24 2188 y Fx(where)41 b(\()p Fv(!)364 2203 y FB(s)400 2188 y Fx(\))438 2203 y FB(s)p Fq(\025)p Ft(0)604 2188 y Fw(2)e Fx(\012)786 2123 y Ft(^)779 2140 y FB(\030)779 2215 y(N)847 2188 y Fx(.)63 b(It)39 b(is)h(the)g(sum)g(of)f(p)s(osition)g(incremen)m(ts)i(along)e(the)h (\014rst)f(co)s(ordinate)h(axis)g(for)-24 2337 y(all)34 b(jumps)h(o)s(ccurring)g(in)f(the)g(time)h(in)m(terv)-5 b(al)35 b([0)p Fv(;)17 b(t)p Fx(].)47 b(Clearly)-8 b(,)36 b Fv(X)2405 2287 y Ft(\(1\))2501 2269 y(^)2494 2287 y FB(\030)2397 2364 y(N)s(;t)2567 2337 y Fx(giv)m(es)f(rise)g(to)f(a)f (time-co)m(v)-5 b(arian)m(t)35 b(and)-24 2458 y(an)m(ti-symmetric)k (family)e(so)g(that,)g(as)g(in)f(Section)i(4.5,)f([DF)m(GW)o(,)h (Theorem)g(2.2])e(can)h(b)s(e)f(used)i(in)f(order)-24 2578 y(to)c(deduce)h(the)f(follo)m(wing)g(result.)-24 2794 y Fm(Prop)s(osition)38 b(6)49 b Fs(Given)34 b Fv(N)39 b Fw(2)28 b Fu(N)p Fs(,)34 b Fv(N)39 b(>)27 b(r)1582 2809 y FB(c)1617 2794 y Fs(,)35 b(and)1881 2768 y Fx(^)1871 2794 y Fv(\030)d Fw(2)2072 2769 y Fx(^)2040 2794 y Fw(N)1344 3043 y Fx(lim)1346 3105 y FB(t)p Fq("1)1534 2975 y Fx(1)p 1534 3020 49 4 v 1541 3111 a Fv(t)1620 3043 y Fm(E)1694 3089 y Fg(P)1755 3043 y Fp(^)1749 3056 y Fr(\030)1749 3112 y(N)1811 2962 y Fo(\000)1857 3043 y Fx(\()p Fv(X)1984 2992 y Ft(\(1\))2080 2974 y(^)2073 2992 y FB(\030)1976 3070 y(N)s(;t)2112 3043 y Fx(\))2150 3002 y Ft(2)2189 2962 y Fo(\001)2290 3043 y Fx(=)55 b Fv(D)2512 2978 y Ft(^)2505 2995 y FB(\030)2502 3070 y(N)2597 3043 y Fv(;)-24 3337 y Fs(wher)-5 b(e)35 b(the)f(di\013usion)g(c)-5 b(o)g(e\016cient)35 b Fv(D)1363 3272 y Ft(^)1356 3290 y FB(\030)1353 3364 y(N)1455 3337 y Fs(is)f(\014nite)h(and)f(given)g(by)951 3604 y Fv(D)1042 3539 y Ft(^)1035 3556 y FB(\030)1032 3631 y(N)1155 3604 y Fx(=)55 b Fv(m)1378 3539 y Ft(^)1371 3556 y FB(\030)1371 3631 y(N)1438 3523 y Fo(\000)1484 3604 y Fv( )1558 3539 y Ft(^)1551 3556 y FB(\030)1547 3631 y(N)1615 3523 y Fo(\001)1710 3604 y Fw(\000)c Fx(2)1919 3468 y Fo(Z)2019 3494 y Fq(1)1975 3694 y Ft(0)2110 3604 y Fv(dt)28 b Fw(h)p Fv(')2334 3539 y Ft(^)2327 3556 y FB(\030)2327 3631 y(N)2411 3604 y Fv(;)33 b(e)2516 3562 y FB(t)p Fq(L)2596 3517 y Fp(^)2590 3530 y Fr(\030)2590 3586 y(N)2669 3604 y Fv(')2740 3539 y Ft(^)2733 3556 y FB(\030)2733 3631 y(N)2800 3604 y Fw(i)2839 3650 y FB(m)2907 3604 y Fp(^)2901 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Fv(x;)17 b(y)t Fw(g)p Fx(\))27 b Fv(d)3327 4113 y Ft(1)3366 4098 y Fx(\()p Fv(x;)17 b(y)t Fx(\))26 b Fv(:)173 b Fx(\(65\))-24 4649 y Fl(6.2)136 b(Link)44 b(to)i(p)t(erio)t(dized)f(medium)-24 4834 y Fx(Here)35 b(w)m(e)g(sho)m(w)g(that)f(the)h(di\013usion)g (matrix)f(\(60\))g(can)g(b)s(e)g(b)s(ounded)h(b)s(elo)m(w)g(in)f(terms) h(of)f(the)g(a)m(v)m(erage)i(of)-24 4954 y(the)27 b(di\013usion)h(co)s (e\016cien)m(t)g(asso)s(ciated)g(to)e(the)h(p)s(erio)s(dized)h(random)f (media.)42 b(Our)27 b(pro)s(of)e(follo)m(ws)j(the)f(argu-)-24 5074 y(men)m(ts)j(of)d([DF)m(GW)o(,)i(Prop.)42 b(4.13],)28 b(but)g(additional)g(tec)m(hnical)h(problems)g(are)f(related)g(to)f (the)h(randomness)-24 5195 y(of)37 b(geometry)h(\(absence)g(of)f(an)m (y)h(lattice)f(structure\))i(and)e(p)s(ossible)h(\(alb)s(eit)f(in)m (tegrable\))h(singularities)h(of)-24 5315 y(the)33 b(mean)h(forw)m(ard) e(v)m(elo)s(cit)m(y)j(and)d(in\014nitesimal)j(mean)e(square)g (displacemen)m(t.)1935 5564 y(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop -24 171 a Fm(Prop)s(osition)38 b(7)49 b Fs(Supp)-5 b(ose)34 b(that)h(for)g Fx(1)27 b Fw(\024)h Fv(p)g Fw(\024)g Fx(8)1003 440 y(lim)986 503 y FB(N)7 b Fq("1)1271 373 y Fv(\032)1321 388 y FB(c)1373 373 y Fv(`)p Fx(\()p Fv(C)1522 388 y Ft(2)p FB(N)1624 373 y Fx(\))p 1182 417 570 4 v 1192 498 a(^)1182 525 y Fv(\030)t Fx(\()p Fv(C)1337 540 y Ft(2)p FB(N)1439 525 y Fx(\))22 b(+)g Fv(a)1648 540 y Ft(2)p FB(N)1816 440 y Fx(=)56 b(1)27 b Fv(;)251 b Fs(in)90 b Fv(L)2543 399 y FB(p)2583 440 y Fx(\()2670 415 y(^)2638 440 y Fw(N)14 b Fv(;)2800 415 y Fx(^)2778 440 y Fw(P)2855 399 y FB(c)2890 440 y Fx(\))27 b Fv(;)837 b Fx(\(66\))-24 753 y Fs(wher)-5 b(e)35 b Fv(\032)302 768 y FB(c)364 753 y Fx(:=)28 b Fm(E)585 765 y Ft(^)569 781 y Fq(P)624 763 y Fr(c)660 753 y Fx(\()708 727 y(^)698 753 y Fv(\030)t Fx(\()p Fv(C)853 768 y Ft(1)892 753 y Fx(\)\))35 b Fs(and)f Fv(a)1243 768 y Ft(2)p FB(N)1374 753 y Fx(:=)27 b Fw(j)p Fx(\000)1593 712 y Fq(\006)1593 780 y FB(N)1660 753 y Fw(j)g Fx(=)h(\(2)p Fv(N)k Fw(\000)23 b Fx(1\))2203 717 y FB(d)p Fq(\000)p Ft(1)2333 753 y Fs(.)45 b(Then,)34 b(for)h(any)f Fv(t)28 b(>)g Fx(0)p Fs(,)893 1004 y Fx(lim)876 1067 y FB(N)7 b Fq("1)1062 1004 y Fm(E)1152 1016 y Ft(^)1136 1033 y Fq(P)1191 1014 y Fr(c)1226 893 y Fo(\020)1286 1004 y Fv(m)1378 939 y Ft(^)1371 957 y FB(\030)1371 1031 y(N)1438 923 y Fo(\000)1484 1004 y Fv( )1558 939 y Ft(^)1551 957 y FB(\030)1547 1031 y(N)1615 923 y Fo(\001)1660 893 y(\021)1775 1004 y Fx(=)55 b Fm(E)1996 1016 y Ft(^)1980 1033 y Fq(P)2035 1010 y Fr(c)2029 1054 y Fp(0)2071 923 y Fo(\000)2117 1004 y Fv( )2184 963 y Ft(\(11\))2180 1029 y FB(c)2313 923 y Fo(\001)2387 1004 y Fv(;)1405 b Fx(\(67\))893 1244 y(lim)876 1307 y FB(N)7 b Fq("1)1062 1244 y Fm(E)1152 1256 y Ft(^)1136 1273 y Fq(P)1191 1254 y Fr(c)1243 1134 y Fo(\020)1302 1244 y Fw(h)p Fv(')1412 1179 y Ft(^)1405 1197 y FB(\030)1405 1271 y(N)1489 1244 y Fv(;)33 b(e)1594 1203 y FB(t)p Fq(L)1674 1157 y Fp(^)1668 1170 y Fr(\030)1668 1226 y(N)1747 1244 y Fv(')1818 1179 y Ft(^)1811 1197 y FB(\030)1811 1271 y(N)1878 1244 y Fw(i)1917 1290 y FB(m)1985 1245 y Fp(^)1979 1258 y Fr(\030)1979 1313 y(N)2041 1134 y Fo(\021)2156 1244 y Fx(=)2287 1163 y Fo(\012)2334 1244 y Fv(')2398 1203 y Ft(\(1\))2398 1269 y FB(c)2509 1244 y Fv(;)g(e)2614 1203 y FB(t)p Fq(L)2688 1211 y Fr(c)2741 1244 y Fv(')2805 1203 y Ft(\(1\))2805 1269 y FB(c)2899 1163 y Fo(\013)2962 1266 y Ft(^)2947 1283 y Fq(P)3002 1260 y Fr(c)2996 1304 y Fp(0)3082 1244 y Fv(;)710 b Fx(\(68\))-24 1538 y Fs(wher)-5 b(e)32 b Fv( )316 1488 y Ft(\(11\))312 1549 y FB(c)478 1538 y Fs(and)g Fv(')729 1488 y Ft(\(1\))729 1549 y FB(c)856 1538 y Fs(ar)-5 b(e)32 b(the)g(\014rst)h(diagonal)e (matrix)h(element)g(of)g(the)h(matrix)f Fv( )3079 1553 y FB(c)3146 1538 y Fs(and)g(the)h(\014rst)f(c)-5 b(omp)g(o-)-24 1659 y(nent)35 b(of)g(the)g(ve)-5 b(ctor)34 b Fv(')826 1674 y FB(c)896 1659 y Fs(intr)-5 b(o)g(duc)g(e)g(d)34 b(in)h Fx(\(59\))p Fs(,)f(r)-5 b(esp)g(e)g(ctively.)123 1846 y Fx(Since)40 b Fv(D)465 1861 y FB(c)539 1846 y Fx(is)f(giv)m(en)i(b)m(y)f(\(60\))e(and)h(is)h(a)f(m)m(ultiple)i(of)d (the)i(iden)m(tit)m(y)h(\()p Fs(cf.)62 b Fx(Remark)40 b(4\),)h(the)e(iden)m(tities)-24 1966 y(\(67\))32 b(and)h(\(68\))f(com) m(bined)i(with)f(F)-8 b(atou's)33 b(Lemma)g(immediately)h(imply:)-24 2186 y Fm(Corollary)k(1)49 b Fs(Under)35 b(the)g(same)f(hyp)-5 b(othesis)34 b(as)h(ab)-5 b(ove,)1154 2459 y Fv(D)1235 2474 y FB(c)1270 2459 y Fx(\()p Fv(r)1352 2474 y FB(c)1387 2459 y Fv(;)17 b(E)1503 2474 y FB(c)1537 2459 y Fx(\))56 b Fw(\025)1763 2319 y Fo(\022)1837 2459 y Fx(lim)17 b(sup)1902 2541 y FB(N)7 b Fq("1)2152 2459 y Fm(E)2242 2471 y Ft(^)2226 2488 y Fq(P)2281 2469 y Fr(c)2317 2378 y Fo(\000)2363 2459 y Fv(D)2454 2394 y Ft(^)2447 2412 y FB(\030)2444 2486 y(N)2511 2378 y Fo(\001)2556 2319 y(\023)2663 2459 y Fm(1)2719 2474 y FB(d)2787 2459 y Fv(;)1005 b Fx(\(69\))-24 2696 y Fs(wher)-5 b(e)35 b Fm(1)308 2711 y FB(d)383 2696 y Fs(is)g(the)f Fv(d)22 b Fw(\002)h Fv(d)34 b Fs(identity)i(matrix.)123 2930 y Fx(Before)g(giving)g(the)g(pro)s(of,)g(let)h(us)f(commen)m(t)h (on)f(its)h(assumptions.)55 b(Due)36 b(to)f(\(57\),)i Fv(\032)3376 2945 y FB(p)3449 2930 y Fv(<)c Fw(1)j Fx(implies)-24 3061 y Fm(E)66 3073 y Ft(^)50 3090 y Fq(P)105 3071 y Fr(c)141 3061 y Fx(\()206 3035 y(^)196 3061 y Fv(\030)t Fx(\()p Fv(C)351 3076 y Ft(1)390 3061 y Fx(\))428 3025 y FB(p)484 3061 y Fx(\))28 b Fv(<)h Fw(1)j Fx(for)g(an)m(y)i Fv(p)28 b(>)g Fx(0.)44 b(As)1590 3036 y(^)1567 3061 y Fw(P)1644 3025 y FB(c)1712 3061 y Fx(is)33 b(ergo)s(dic,)h(this)f (implies)i(the)e(follo)m(wing)g(ergo)s(dic)g(theorem,)-24 3182 y(an)h(extension)i(of)d([D)m(V,)h(Theorem)h(10.2].)47 b(W)-8 b(e)34 b(recall)g(that)g(a)f(con)m(v)m(ex)j(a)m(v)m(eraging)f (sequence)h(of)d(sets)i Fw(f)p Fv(A)3895 3197 y FB(n)3942 3182 y Fw(g)-24 3302 y Fx(in)k Fu(R)168 3266 y FB(d)248 3302 y Fx(is)g(a)f(sequence)k(of)c(con)m(v)m(ex)j(sets)f(suc)m(h)h (that)d Fv(A)2007 3317 y FB(n)2093 3302 y Fw(\032)g Fv(A)2281 3317 y FB(n)p Ft(+1)2457 3302 y Fx(and)h Fv(A)2726 3317 y FB(n)2812 3302 y Fx(con)m(tains)h(a)f(ball)f(of)h(radius)g Fv(r)3945 3317 y FB(n)-24 3422 y Fx(with)34 b Fv(r)243 3437 y FB(n)317 3422 y Fw(!)27 b(1)33 b Fx(as)f Fv(n)c Fw(!)g(1)p Fx(.)-24 3642 y Fm(Lemma)39 b(6)49 b Fs(Supp)-5 b(ose)30 b(that)h Fv(\032)1090 3657 y FB(p)1158 3642 y Fv(<)d Fw(1)p Fs(,)j Fv(p)c Fw(\025)h Fx(1)p Fs(.)44 b(Then,)30 b(given)g(a)h(c)-5 b(onvex)29 b(aver)-5 b(aging)30 b(se)-5 b(quenc)g(e)30 b(of)g(Bor)-5 b(el)30 b(sets)-24 3763 y Fw(f)p Fv(A)99 3778 y FB(n)146 3763 y Fw(g)35 b Fs(in)g Fu(R)423 3727 y FB(d)463 3763 y Fs(,)1303 3865 y Fx(^)1293 3891 y Fv(\030)5 b Fx(\()p Fv(A)1452 3906 y FB(n)1499 3891 y Fx(\))p 1246 3936 339 4 v 1246 4027 a Fv(\032)1296 4042 y FB(c)1347 4027 y Fv(`)p Fx(\()p Fv(A)1499 4042 y FB(n)1546 4027 y Fx(\))1649 3959 y Fw(!)55 b Fx(1)199 b Fs(in)90 b Fv(L)2293 3918 y FB(p)2333 3959 y Fx(\()2420 3934 y(^)2388 3959 y Fw(N)14 b Fv(;)2551 3934 y Fx(^)2528 3959 y Fw(P)2605 3918 y FB(c)2640 3959 y Fx(\))28 b Fv(;)-24 4184 y Fs(and)1468 4261 y Fx(^)1458 4287 y Fv(\030)t Fx(\()p Fv(A)1616 4302 y FB(n)1663 4287 y Fx(\))p 1410 4332 V 1410 4423 a Fv(\032)1460 4438 y FB(c)1512 4423 y Fv(`)p Fx(\()p Fv(A)1664 4438 y FB(n)1711 4423 y Fx(\))1814 4355 y Fw(!)55 b Fx(1)2240 4330 y(^)2217 4355 y Fw(P)2294 4314 y FB(c)2329 4355 y Fs(-a.s.)27 b Fv(:)123 4618 y Fx(W)-8 b(e)36 b(will)h(also)f(need)h(a)e(b)s(ound)h (on)g Fm(E)1529 4630 y Ft(^)1513 4647 y Fq(P)1568 4628 y Fr(c)1604 4618 y Fx(\(\()1690 4592 y(^)1680 4618 y Fv(\030)t Fx(\()p Fv(A)1838 4633 y FB(n)1885 4618 y Fx(\))p Fv(=`)p Fx(\()p Fv(A)2124 4633 y FB(n)2171 4618 y Fx(\)\))2247 4582 y FB(p)2286 4618 y Fx(\),)h(uniformly)g(in)f Fv(n)p Fx(,)h(for)e(a)h(sequence)i(of)e(sets)-24 4739 y(that)31 b(do)s(es)g(not)g(satisfy)g(the)h(assumptions)g(of)e(Lemma)i(6.)42 b(T)-8 b(o)31 b(this)h(aim)e(w)m(e)i(note)f(that,)g(giv)m(en)h(a)e (Borel)h(set)-24 4859 y Fv(B)i Fw(\032)28 b Fu(R)260 4823 y FB(d)333 4859 y Fx(whic)m(h)34 b(is)f(a)g(union)g(of)f Fv(k)j Fx(non-o)m(v)m(erlapping)f(cub)s(es)g(of)e(side)i(1,)e(one)h (has)848 5151 y Fm(E)938 5163 y Ft(^)922 5179 y Fq(P)977 5160 y Fr(c)1012 5070 y Fo(\000)16 b(\000)1140 5057 y Fx(^)1130 5083 y Fv(\030)t Fx(\()p Fv(B)5 b Fx(\))p 1130 5128 203 4 v 1204 5219 a Fv(k)1342 5070 y Fo(\001)1388 5092 y FB(p)1444 5070 y Fo(\001)1545 5151 y Fw(\024)56 b Fm(E)1768 5163 y Ft(^)1752 5179 y Fq(P)1807 5160 y Fr(c)1843 5070 y Fo(\000)1915 5124 y Fx(^)1905 5151 y Fv(\030)5 b Fx(\()p 1991 5070 77 4 v Fv(C)2067 5166 y Ft(1)2107 5151 y Fx(\))2145 5109 y FB(p)2184 5070 y Fo(\001)2285 5151 y Fw(\024)56 b Fv(\032)2468 5166 y FB(p)2525 5151 y Fv(;)212 b Fw(8)28 b Fv(p)f Fw(\025)h Fx(1)17 b Fv(;)698 b Fx(\(70\))1935 5564 y(30)p eop end %%Page: 31 31 TeXDict begin 31 30 bop -24 171 a Fx(where)p 253 91 77 4 v 29 w Fv(C)330 186 y Ft(1)397 171 y Fx(is)28 b(the)g(closed)h(cub)s (e)f([)p Fw(\000)1275 131 y Ft(1)p 1275 148 36 4 v 1275 205 a(2)1321 171 y Fv(;)1374 131 y Ft(1)p 1374 148 V 1374 205 a(2)1419 171 y Fx(])1446 134 y FB(d)1487 171 y Fx(.)42 b(This)28 b(follo)m(ws)h(from)e(the)h(stationarit)m(y)g(of) 3127 145 y(^)3104 171 y Fw(P)3181 134 y FB(c)3244 171 y Fx(and)f(the)h(con)m(v)m(exit)m(y)-24 291 y(of)k(the)h(function)h Fv(f)11 b Fx(\()p Fv(x)p Fx(\))27 b(=)h Fv(x)1014 255 y FB(p)1054 291 y Fx(,)33 b Fv(x)28 b Fw(\025)g Fx(0.)-24 459 y Fm(Pro)s(of)44 b(of)f(Prop)s(osition)h(7)p Fx(.)59 b(Without)38 b(loss)h(of)e(generalit)m(y)-8 b(,)41 b(w)m(e)e(assume)g Fv(r)2895 474 y FB(c)2967 459 y Fx(=)d(1.)59 b(A)38 b(k)m(ey)i(observ) -5 b(ation)-24 579 y(in)34 b(order)g(to)g(pro)m(v)m(e)h(\(67\))e(and)h (\(68\))f(is)h(the)g(follo)m(wing)g(iden)m(tit)m(y)-8 b(,)36 b(v)-5 b(alid)34 b(for)f(an)m(y)i(nonnegativ)m(e)g(measurable) -24 699 y(function)e Fv(h)g Fx(de\014ned)h(on)951 674 y(^)919 699 y Fw(N)1001 714 y Ft(0)1040 699 y Fx(.)43 b(It)33 b(follo)m(ws)g(easily)h(from)e(\(7\):)682 977 y Fm(E)772 989 y Ft(^)756 1005 y Fq(P)811 986 y Fr(c)863 836 y Fo(\022)936 841 y(Z)992 1067 y FB(B)1079 950 y Fx(^)1069 977 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))p Fv(h)p Fx(\()p Fv(S)1452 992 y FB(x)1507 950 y Fx(^)1496 977 y Fv(\030)5 b Fx(\))1582 836 y Fo(\023)1710 977 y Fx(=)56 b Fv(\032)1892 992 y FB(c)1943 977 y Fv(`)p Fx(\()p Fv(B)5 b Fx(\))17 b Fm(E)2246 989 y Ft(^)2230 1005 y Fq(P)2285 982 y Fr(c)2279 1026 y Fp(0)2320 977 y Fx(\()p Fv(h)p Fx(\))p Fv(;)212 b Fw(8)56 b Fv(B)33 b Fw(2)28 b(B)s Fx(\()p Fu(R)3181 936 y FB(d)3222 977 y Fx(\))p Fv(:)532 b Fx(\(71\))-24 1274 y(F)-8 b(rom)32 b(this)i(iden)m(tit)m(y)g(w)m(e)g (can)e(deduce)j(for)d(an)m(y)h Fv(h)28 b Fw(2)g Fv(L)2008 1237 y Ft(2)2048 1274 y Fx(\()p Fw(N)2168 1289 y Ft(0)2207 1274 y Fv(;)2273 1248 y Fx(^)2251 1274 y Fw(P)2328 1237 y FB(c)2320 1298 y Ft(0)2363 1274 y Fx(\))k(that)764 1581 y(lim)747 1644 y FB(N)7 b Fq("1)933 1581 y Fm(E)1023 1593 y Ft(^)1007 1609 y Fq(P)1062 1590 y Fr(c)1114 1410 y Fo( )1463 1513 y Fx(1)p 1203 1558 570 4 v 1213 1639 a(^)1203 1665 y Fv(\030)t Fx(\()p Fv(C)1358 1680 y Ft(2)p FB(N)1460 1665 y Fx(\))23 b(+)f Fv(a)1670 1680 y Ft(2)p FB(N)1799 1445 y Fo(Z)1854 1671 y FB(C)1904 1682 y Fp(2)p Fr(N)5 b Fk(\000)p Fp(2)2102 1554 y Fx(^)2092 1581 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))p Fv(h)p Fx(\()p Fv(S)2475 1596 y FB(x)2529 1554 y Fx(^)2519 1581 y Fv(\030)g Fx(\))2605 1410 y Fo(!)2739 1581 y Fx(=)55 b Fm(E)2960 1593 y Ft(^)2944 1609 y Fq(P)2999 1586 y Fr(c)2993 1631 y Fp(0)3035 1581 y Fx(\()p Fv(h)p Fx(\))27 b Fv(:)598 b Fx(\(72\))-24 1888 y(In)33 b(fact,)g(due)g(to)f(\(71\),)g(it)h(is)g(enough)g(to)f (sho)m(w)i(that)114 2195 y Fm(E)204 2207 y Ft(^)188 2224 y Fq(P)243 2205 y Fr(c)295 2025 y Fo( )390 2085 y(\020)720 2128 y Fx(1)p 460 2172 V 470 2253 a(^)460 2280 y Fv(\030)t Fx(\()p Fv(C)615 2295 y Ft(2)p FB(N)717 2280 y Fx(\))22 b(+)g Fv(a)926 2295 y Ft(2)p FB(N)1061 2195 y Fw(\000)1379 2128 y Fx(1)p 1171 2172 465 4 v 1171 2263 a Fv(\032)1221 2278 y FB(c)1256 2263 y Fv(`)p Fx(\()p Fv(C)1405 2278 y Ft(2)p FB(N)7 b Fq(\000)p Ft(2)1597 2263 y Fx(\))1645 2085 y Fo(\021)1738 2060 y(Z)1793 2285 y FB(C)1843 2296 y Fp(2)p Fr(N)e Fk(\000)p Fp(2)2041 2169 y Fx(^)2031 2195 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))p Fv(h)p Fx(\()p Fv(S)2414 2210 y FB(x)2468 2169 y Fx(^)2458 2195 y Fv(\030)g Fx(\))2544 2025 y Fo(!)2678 2195 y Fw(#)55 b Fx(0)27 b Fv(;)245 b Fx(as)60 b Fv(N)66 b Fw(")55 b(1)27 b Fv(:)138 b Fx(\(73\))-24 2502 y(By)33 b(applying)h(t)m(wice)g(the)f(Cauc)m(h)m (y-Sc)m(h)m(w)m(arz)j(inequalit)m(y)f(and)d(b)m(y)i(in)m(v)m(oking)g (\(71\),)e(w)m(e)i(obtain)179 2656 y Fo(\020)238 2766 y Fx(l.h.s.)45 b(of)32 b(\(73\))768 2656 y Fo(\021)827 2678 y Ft(2)262 3039 y Fw(\024)56 b Fm(E)485 3051 y Ft(^)469 3067 y Fq(P)524 3048 y Fr(c)576 2868 y Fo( )671 2928 y(\020)785 2971 y Fv(\032)835 2986 y FB(c)887 2971 y Fv(`)p Fx(\()p Fv(C)1036 2986 y Ft(2)p FB(N)7 b Fq(\000)p Ft(2)1228 2971 y Fx(\))p 741 3016 570 4 v 751 3097 a(^)741 3123 y Fv(\030)t Fx(\()p Fv(C)896 3138 y Ft(2)p FB(N)999 3123 y Fx(\))22 b(+)g Fv(a)1208 3138 y Ft(2)p FB(N)1342 3039 y Fw(\000)h Fx(1)1491 2928 y Fo(\021)1550 2950 y Ft(2)1680 2945 y Fx(^)1669 2971 y Fv(\030)5 b Fx(\()p Fv(C)1825 2986 y Ft(2)p FB(N)i Fq(\000)p Ft(2)2017 2971 y Fx(\))p 1600 3016 526 4 v 1600 3107 a Fv(\032)1650 3078 y Ft(2)1650 3132 y FB(c)1706 3107 y Fv(`)p Fx(\()p Fv(C)1855 3122 y Ft(2)p FB(N)g Fq(\000)p Ft(2)2047 3107 y Fx(\))2085 3078 y Ft(2)2135 2868 y Fo(!)2230 3039 y Fm(E)2320 3051 y Ft(^)2304 3067 y Fq(P)2359 3048 y Fr(c)2412 2868 y Fo( )2669 2971 y Fx(1)p 2500 3016 386 4 v 2510 3097 a(^)2500 3123 y Fv(\030)e Fx(\()p Fv(C)2656 3138 y Ft(2)p FB(N)i Fq(\000)p Ft(2)2848 3123 y Fx(\))2896 2928 y Fo(\020)2956 2903 y(Z)3011 3129 y FB(C)3061 3140 y Fp(2)p Fr(N)e Fk(\000)p Fp(2)3259 3012 y Fx(^)3249 3039 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))p Fv(h)p Fx(\()p Fv(S)3632 3054 y FB(x)3686 3012 y Fx(^)3676 3039 y Fv(\030)t Fx(\))3761 2928 y Fo(\021)3821 2950 y Ft(2)3860 2868 y Fo(!)262 3371 y Fw(\024)56 b Fm(E)485 3383 y Ft(^)469 3399 y Fq(P)524 3380 y Fr(c)576 3200 y Fo( )671 3260 y(\020)785 3303 y Fv(\032)835 3318 y FB(c)887 3303 y Fv(`)p 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b Fx(=)h(4.)123 3942 y(Let)k(no)m(w)i Fv(h)564 3878 y Ft(^)557 3895 y FB(\030)557 3969 y(N)657 3942 y Fx(b)s(e)e(a)h(function)g(on)f Fw(V)1464 3878 y Ft(^)1457 3895 y FB(\030)1449 3969 y(N)1549 3942 y Fx(suc)m(h)i(that)e(for)g(some)i(constan)m(t)f Fv(c)28 b(>)g Fx(0)k(indep)s(enden)m(t)j(of)d Fv(N)1163 4281 y Fw(j)p Fv(h)1254 4217 y Ft(^)1247 4234 y FB(\030)1247 4308 y(N)1314 4281 y Fx(\()p Fv(x)p Fx(\))p Fw(j)55 b(\024)h Fv(c)1748 4077 y Fo(8)1748 4167 y(<)1748 4346 y(:)1846 4174 y Fx(^)1836 4200 y Fv(\030)5 b Fx(\()p 1922 4114 245 4 v Fv(B)1996 4215 y Ft(1)2035 4200 y Fx(\()p Fv(x)p Fx(\)\))130 b(if)33 b Fv(x)28 b Fw(2)g Fv(Q)2685 4136 y Ft(^)2678 4153 y FB(\030)2678 4227 y(N)2745 4200 y Fv(;)1846 4324 y Fq(j)p FB(B)1928 4279 y Fp(^)1922 4292 y Fr(\030)1919 4347 y(N)1977 4324 y Fq(j)p 1846 4355 151 4 v 1858 4412 a FB(a)1895 4423 y Fp(2)p Fr(N)2334 4378 y Fx(otherwise)34 b Fv(;)-24 4655 y Fx(where)g Fv(B)344 4590 y Ft(^)337 4607 y FB(\030)332 4682 y(N)427 4655 y Fx(=)28 b Fv(B)617 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Fr(\030)1641 5209 y(N)1698 5186 y Fq(n)p FB(Q)1795 5140 y Fp(^)1789 5153 y Fr(\030)1789 5209 y(N)5 b Fk(\000)p Fp(1)1942 5034 y Fw(j)p Fv(h)2033 4969 y Ft(^)2026 4987 y FB(\030)2026 5061 y(N)2093 5034 y Fx(\()p Fv(x)p Fx(\))p Fw(j)2252 4993 y FB(p)2292 4923 y Fo(\021)2406 5034 y Fx(=)56 b(0)27 b Fv(;)244 b Fx(for)33 b(1)27 b Fw(\024)h Fv(p)g Fw(\024)g Fx(4)p Fv(:)345 b Fx(\(74\))1935 5564 y(31)p eop end %%Page: 32 32 TeXDict begin 32 31 bop -24 171 a Fx(In)33 b(fact,)g(the)g(sum)g(in)g (\(74\))f(can)h(b)s(e)g(b)s(ounded)g(b)m(y)1132 478 y Fv(c)1174 437 y FB(p)1230 478 y Fv(a)1281 493 y Ft(2)p FB(N)1394 410 y Fw(j)p Fv(B)1508 346 y Ft(^)1501 363 y FB(\030)1496 437 y(N)1563 410 y Fw(j)1591 374 y FB(p)p 1394 455 237 4 v 1435 546 a Fv(a)1486 499 y FB(p)1486 573 y Ft(2)p FB(N)1690 478 y Fx(+)50 b Fv(c)1858 437 y FB(p)2072 383 y Fo(X)1931 630 y FB(x)p Fq(2)p FB(Q)2080 584 y Fp(^)2074 597 y Fr(\030)2074 653 y(N)2131 630 y Fq(n)p FB(Q)2228 584 y Fp(^)2222 597 y Fr(\030)2222 653 y(N)5 b Fk(\000)p Fp(1)2384 452 y Fx(^)2374 478 y Fv(\030)t Fx(\()p 2459 391 245 4 v Fv(B)2533 493 y Ft(1)2573 478 y Fx(\()p Fv(x)p Fx(\)\))2742 437 y FB(p)2810 478 y Fv(:)982 b Fx(\(75\))-24 843 y(By)33 b(the)g(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)k (inequalit)m(y)494 1145 y Fm(E)584 1157 y Ft(^)568 1173 y Fq(P)623 1154 y Fr(c)659 1034 y Fo(\020)936 1077 y Fv(a)987 1092 y Ft(2)p FB(N)p 728 1122 570 4 v 738 1203 a Fx(^)728 1229 y Fv(\030)5 b Fx(\()p Fv(C)884 1244 y Ft(2)p FB(N)986 1229 y Fx(\))22 b(+)g Fv(a)1195 1244 y Ft(2)p FB(N)1345 1077 y Fw(j)p Fv(B)1459 1012 y Ft(^)1452 1030 y FB(\030)1447 1104 y(N)1514 1077 y Fw(j)1542 1041 y FB(p)p 1345 1122 237 4 v 1387 1213 a Fv(a)1438 1166 y FB(p)1438 1240 y Ft(2)p FB(N)1592 1034 y Fo(\021)1707 1145 y Fw(\024)55 b Fm(E)1923 1057 y Fp(1)p 1923 1069 31 3 v 1923 1110 a(2)1929 1172 y Ft(^)1913 1189 y Fq(P)1968 1170 y Fr(c)2004 1034 y Fo(\020)2339 1077 y Fv(a)2390 1041 y Ft(2)2390 1102 y(2)p FB(N)p 2073 1122 685 4 v 2073 1229 a Fx(\()2121 1203 y(^)2111 1229 y 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2820 y FB(c)1142 2805 y Fv(`)p Fx(\()p Fv(C)1291 2820 y Ft(2)p FB(N)1415 2805 y Fw(n)22 b Fv(C)1557 2820 y Ft(2)p FB(N)7 b Fq(\000)p Ft(2)1749 2805 y Fx(\))p Fm(E)1877 2817 y Ft(^)1861 2834 y Fq(P)1916 2815 y Fr(c)1952 2695 y Fo(\020)2048 2712 y Fx(^)2038 2738 y Fv(\030)t Fx(\()p Fv(C)2193 2753 y Ft(2)p FB(N)2318 2738 y Fw(n)22 b Fv(C)2460 2753 y Ft(2)p FB(N)7 b Fq(\000)p Ft(2)2652 2738 y Fx(\))p 2021 2782 685 4 v 2021 2890 a(\()2069 2864 y(^)2059 2890 y Fv(\030)e Fx(\()p Fv(C)2215 2905 y Ft(2)p FB(N)2317 2890 y Fx(\))22 b(+)g Fv(a)2526 2905 y Ft(2)p FB(N)2629 2890 y Fx(\))2667 2861 y Ft(2)2716 2695 y Fo(\021)17 b(\021)2862 2690 y Fp(1)p 2862 2702 31 3 v 2862 2743 a(2)2934 2805 y Fm(E)3018 2718 y Fp(1)p 3018 2730 V 3018 2771 a(2)3023 2833 y Ft(^)3008 2850 y Fq(P)3063 2831 y Fr(c)3099 2695 y Fo(\020)3168 2779 y Fx(^)3158 2805 y Fv(\030)5 b Fx(\()p 3244 2719 239 4 v Fv(B)3318 2820 y Ft(1)3357 2805 y Fx(\(0\)\))3520 2764 y Ft(2)p FB(p)3595 2695 y Fo(\021)3682 2805 y Fv(:)-24 3088 y Fx(The)41 b(last)f(factor)f(is)i(b)s(ounded)f(b)m(y)h(h)m(yp)s (othesis,)j(the)c(\014rst)g(one)g(con)m(v)m(erges)i(to)d(0)h(as)g Fv(N)50 b Fw(")40 b(1)f Fx(b)s(ecause)j(of)-24 3208 y(Lemma)34 b(6)e(and)h(\(66\).)123 3400 y(In)28 b(order)g(to)g(pro)m(v)m(e)h (\(67\))f(observ)m(e)h(that)f Fv( )1681 3349 y Ft(\(11\))1677 3410 y FB(c)1811 3400 y Fx(\()p Fv(S)1909 3415 y FB(x)1963 3373 y Fx(^)1953 3400 y Fv(\030)t Fx(\))g(=)f Fv( )2243 3335 y Ft(^)2236 3353 y FB(\030)2232 3427 y(N)2300 3400 y Fx(\()p Fv(x)p Fx(\))h(if)g Fv(x)g Fw(2)g Fv(Q)2805 3335 y Ft(^)2798 3353 y FB(\030)2798 3427 y(N)7 b Fq(\000)p Ft(1)2956 3400 y Fx(.)42 b(Therefore)29 b(w)m(e)g(can)f(write)158 3673 y Fv(m)250 3609 y Ft(^)243 3626 y FB(\030)243 3700 y(N)310 3673 y Fx(\()p Fv( )422 3609 y Ft(^)415 3626 y FB(\030)411 3700 y(N)479 3673 y Fx(\))55 b(=)974 3606 y(1)p 713 3650 570 4 v 723 3732 a(^)713 3758 y Fv(\030)5 b Fx(\()p Fv(C)869 3773 y Ft(2)p FB(N)971 3758 y Fx(\))22 b(+)g Fv(a)1180 3773 y Ft(2)p FB(N)1310 3538 y Fo(Z)1365 3763 y FB(C)1415 3774 y Fp(2)p Fr(N)5 b Fk(\000)p Fp(2)1613 3647 y Fx(^)1602 3673 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))p Fv( )1899 3632 y Ft(\(11\))1895 3698 y FB(c)2028 3673 y Fx(\()p Fv(S)2126 3688 y FB(x)2180 3647 y Fx(^)2170 3673 y Fv(\030)g Fx(\))49 b(+)2701 3606 y(1)p 2441 3650 V 2451 3732 a(^)2441 3758 y Fv(\030)5 b Fx(\()p Fv(C)2597 3773 y Ft(2)p FB(N)2699 3758 y Fx(\))22 b(+)g Fv(a)2908 3773 y Ft(2)p FB(N)3185 3579 y Fo(X)3037 3825 y FB(x)p Fq(2V)3179 3780 y Fp(^)3174 3793 y Fr(\030)3168 3848 y(N)3225 3825 y Fq(n)p FB(C)3310 3836 y Fp(2)p Fr(N)5 b Fk(\000)p Fp(2)3494 3673 y Fv( )3568 3609 y Ft(^)3561 3626 y FB(\030)3557 3700 y(N)3625 3673 y Fx(\()p Fv(x)p Fx(\))28 b Fv(:)-24 4073 y Fx(No)m(w)35 b(\(67\))e(follo)m(ws)h(easily) h(from)f(\(72\))f(and)h(\(74\))f(with)h Fv(h)2121 4008 y Ft(^)2114 4026 y FB(\030)2114 4100 y(N)2212 4073 y Fx(:=)29 b Fv( )2418 4008 y Ft(^)2411 4026 y FB(\030)2407 4100 y(N)2475 4073 y Fx(.)47 b(Note)34 b(that)f(b)m(y)i(the)f(same)h 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y FB(M)1705 2732 y Fo(\001)1750 2702 y(o)1844 2813 y Fm(E)1928 2725 y Fp(1)p 1928 2737 V 1928 2779 a(2)1934 2841 y Ft(^)1918 2857 y Fq(P)1973 2838 y Fr(c)2009 2702 y Fo(n)2075 2813 y Fv(m)2167 2748 y Ft(^)2160 2766 y FB(\030)2160 2840 y(N)2228 2732 y Fo(\002)2297 2813 y Fv(')2368 2748 y Ft(^)2361 2766 y FB(\030)2361 2840 y(N)2428 2813 y Fx(\()p Fv(x)p Fx(\))2559 2772 y Ft(2)2599 2813 y Fm(E)2673 2859 y Fg(P)2734 2813 y Fp(^)2728 2826 y Fr(\030)2728 2882 y(N)r(;x)2840 2732 y Fo(\000)2886 2813 y Fv(')2957 2748 y Ft(^)2950 2766 y FB(\030)2950 2840 y(N)3017 2813 y Fx(\()p Fv(!)3116 2828 y FB(t)3145 2813 y Fx(\))3183 2772 y Ft(2)3223 2732 y Fo(\001)27 b(\003)3338 2702 y(o)766 3044 y Fw(\024)h Fm(E)955 2957 y Fp(1)p 954 2969 V 954 3010 a(2)960 3072 y Ft(^)945 3089 y Fq(P)1000 3070 y Fr(c)1036 2934 y Fo(n)1102 3044 y Fv(m)1194 2980 y Ft(^)1187 2997 y FB(\030)1187 3071 y(N)1255 2964 y Fo(\000)1317 3044 y Fw(V)1393 2980 y Ft(^)1386 2997 y FB(\030)1378 3071 y(N)1467 3044 y Fw(n)22 b Fv(C)1609 3059 y FB(M)1705 2964 y Fo(\001)1750 2934 y(o)1844 3044 y Fm(E)1928 2957 y Fp(1)p 1928 2969 V 1928 3010 a(4)1934 3072 y Ft(^)1918 3089 y Fq(P)1973 3070 y Fr(c)2009 2934 y Fo(n)2075 3044 y Fv(m)2167 2980 y Ft(^)2160 2997 y FB(\030)2160 3071 y(N)2228 2964 y Fo(\002)2297 3044 y Fv(')2368 2980 y Ft(^)2361 2997 y FB(\030)2361 3071 y(N)2428 3044 y Fx(\()p Fv(x)p Fx(\))2559 3003 y Ft(4)2627 2964 y Fo(\003)2668 2934 y(o)2762 3044 y Fm(E)2846 2957 y Fp(1)p 2846 2969 V 2846 3010 a(4)2852 3072 y Ft(^)2836 3089 y Fq(P)2891 3070 y Fr(c)2927 2934 y Fo(n)2993 3044 y Fm(E)3067 3090 y Fg(P)3128 3045 y Fp(^)3122 3058 y Fr(\030)3122 3114 y(N)3184 2964 y Fo(\000)3230 3044 y Fv(')3301 2980 y Ft(^)3294 2997 y FB(\030)3294 3071 y(N)3361 3044 y Fx(\()p Fv(!)3460 3059 y FB(t)3489 3044 y Fx(\))3527 3003 y Ft(4)3567 2964 y Fo(\001)27 b(\003)3682 2934 y(o)766 3266 y Fx(=)g Fm(E)953 3179 y Fp(1)p 953 3191 V 953 3232 a(2)959 3294 y Ft(^)943 3311 y Fq(P)998 3292 y Fr(c)1034 3156 y Fo(n)1100 3266 y Fv(m)1192 3202 y Ft(^)1185 3219 y FB(\030)1185 3293 y(N)1253 3186 y Fo(\000)1315 3266 y Fw(V)1391 3202 y Ft(^)1384 3219 y FB(\030)1376 3293 y(N)1466 3266 y Fw(n)22 b Fv(C)1608 3281 y FB(M)1703 3186 y Fo(\001)1749 3156 y(o)1843 3266 y Fm(E)1927 3179 y Fp(1)p 1926 3191 V 1926 3232 a(2)1932 3294 y Ft(^)1917 3311 y Fq(P)1972 3292 y Fr(c)2007 3156 y Fo(n)2074 3266 y Fv(m)2166 3202 y Ft(^)2159 3219 y FB(\030)2159 3293 y(N)2226 3186 y Fo(\002)2296 3266 y Fv(')2367 3202 y Ft(^)2360 3219 y FB(\030)2360 3293 y(N)2427 3266 y Fx(\()p Fv(x)p Fx(\))2558 3225 y Ft(4)2625 3186 y Fo(\003)2667 3156 y(o)2761 3266 y Fv(;)-24 3550 y Fx(where)37 b(the)g(last)f(iden)m(tit)m(y)h(follo)m (ws)g(from)e(the)h(stationarit)m(y)h(of)e Fw(L)2432 3485 y Ft(^)2425 3503 y FB(\030)2425 3577 y(N)2528 3550 y Fx(w.r.t.)53 b Fv(m)2900 3485 y Ft(^)2893 3503 y FB(\030)2893 3577 y(N)2961 3550 y Fx(.)g(Due)36 b(to)f(the)h(dominated)-24 3670 y(con)m(v)m(ergence)31 b(theorem,)f(the)f(\014rst)g(exp)s (ectation)g(on)f(the)h(r.h.s.)42 b(go)s(es)29 b(to)f(0,)g(while)i(the)e (second)i(exp)s(ectation)-24 3791 y(is)j(b)s(ounded)h(due)f(to)f (\(76\).)123 3958 y(In)e(order)f(to)h(pro)m(v)m(e)h(the)f(second)g (limit)h(in)e(\(78\),)h(w)m(e)h(apply)f(t)m(wice)h(the)f(Cauc)m(h)m (y-Sc)m(h)m(w)m(arz)j(inequalit)m(y)f(in)-24 4102 y(order)h(to)f (obtain)h(the)g(b)s(ound)g Fm(E)1213 4114 y Ft(^)1197 4131 y Fq(P)1252 4112 y Fr(c)1287 4022 y Fo(\000)1333 4102 y Fv(A)1413 4038 y Ft(^)1406 4055 y FB(\030)1406 4129 y Ft(2)p FB(;N)1528 4022 y Fo(\001)1606 4102 y Fx(b)m(y)166 4370 y Fm(E)250 4282 y Fp(1)p 250 4294 V 250 4336 a(2)255 4398 y Ft(^)240 4414 y Fq(P)295 4396 y Fr(c)331 4259 y Fo(n)397 4370 y Fv(m)489 4305 y Ft(^)482 4323 y FB(\030)482 4397 y(N)550 4289 y Fo(\002)619 4370 y Fv(')690 4305 y Ft(^)683 4323 y FB(\030)683 4397 y(N)750 4370 y Fx(\()p Fv(x)p Fx(\))881 4329 y Ft(2)948 4289 y Fo(\003)990 4259 y(o)1056 4370 y Fm(E)1140 4282 y Fp(1)p 1140 4294 V 1140 4336 a(4)1146 4398 y Ft(^)1130 4414 y Fq(P)1185 4396 y Fr(c)1221 4259 y Fo(n)1287 4370 y Fm(E)1361 4416 y Fg(P)1422 4370 y Fp(^)1416 4383 y Fr(\030)1416 4439 y(N)1478 4289 y Fo(\002)1536 4370 y Fv(')1607 4305 y Ft(^)1600 4323 y FB(\030)1600 4397 y(N)1668 4370 y Fx(\()p Fv(!)1767 4385 y FB(t)1796 4370 y Fx(\))1834 4329 y Ft(4)1901 4289 y Fo(\003)1942 4259 y(o)2009 4370 y Fm(E)2093 4282 y Fp(1)p 2092 4294 V 2092 4336 a(4)2098 4398 y Ft(^)2083 4414 y Fq(P)2138 4396 y Fr(c)2173 4259 y Fo(n)2240 4370 y Fv(m)2332 4305 y Ft(^)2325 4323 y FB(\030)2325 4397 y(N)2392 4289 y Fo(\002)2434 4370 y Fv(\037)p Fx(\()p Fv(x)28 b Fw(2)g Fv(C)2780 4385 y FB(M)2859 4370 y Fx(\))17 b Fm(P)2998 4305 y Ft(^)2991 4323 y FB(\030)2991 4397 y(N)s(;x)3113 4370 y Fx(\()p Fv(\034)3211 4305 y Ft(^)3204 4323 y FB(\030)3193 4397 y(N)3288 4370 y Fw(\024)28 b Fv(t)p Fx(\))3466 4289 y Fo(\003)3508 4259 y(o)3602 4370 y Fv(:)190 b Fx(\(79\))-24 4622 y(Again,)38 b(b)s(ecause)g(of)e (stationarit)m(y)h(and)g(\(76\),)g(the)g(\014rst)g(t)m(w)m(o)g(factors) g(on)g(the)g(r.h.s.)56 b(are)37 b(b)s(ounded)g(while)-24 4742 y(the)c(last)g(one)g(con)m(v)m(erges)i(to)d(0)g(due)i(to)e(Lemma)h (7)f(b)s(elo)m(w.)123 4910 y(Finally)44 b(w)m(e)g(pro)m(v)m(e)h(the)g (last)f(limit)g(in)g(\(78\).)76 b(T)-8 b(o)44 b(this)g(aim,)j(giv)m(en) 2779 4883 y(^)2769 4910 y Fv(\030)p 2769 4945 48 4 v 51 w Fw(2)2984 4885 y Fx(^)2976 4910 y(\004)f(=)h Fv(D)s Fx(\([0)p Fv(;)17 b Fw(1)p Fx(\))p Fv(;)3664 4885 y Fx(^)3634 4910 y Fw(N)3716 4925 y Ft(0)3753 4910 y Fx(\))44 b(and)-24 5030 y Fv(x)28 b Fw(2)h Fv(C)224 5045 y FB(M)302 5030 y Fx(,)k(w)m(e)h(set)1033 5271 y Fv(\034)1075 5286 y FB(N)s(;x)1198 5271 y Fx(\()1246 5245 y(^)1236 5271 y Fv(\030)p 1236 5306 V 4 w Fx(\))28 b(=)g(inf)1587 5160 y Fo(n)1654 5271 y Fv(s)f Fw(\025)h Fx(0)44 b(:)h Fv(x)22 b Fx(+)g Fv(X)2253 5286 y FB(s)2290 5271 y Fx(\()2338 5245 y(^)2328 5271 y Fv(\030)p 2328 5306 V 5 w Fx(\))27 b Fw(62)h Fv(C)2605 5286 y Ft(2)p FB(N)7 b Fq(\000)p Ft(2)2798 5160 y Fo(o)2908 5271 y Fv(;)884 b Fx(\(80\))1935 5564 y(33)p eop end %%Page: 34 34 TeXDict begin 34 33 bop -24 171 a Fx(where)34 b Fv(X)339 186 y FB(s)376 171 y Fx(\()424 144 y(^)414 171 y Fv(\030)p 414 206 48 4 v 4 w Fx(\))f(is)g(de\014ned)h(as)f(in)f(\(46\).)43 b(Note)33 b(that)f(for)g Fv(x)d Fw(2)f Fv(C)2287 186 y FB(M)2388 171 y Fw(\\)22 b Fx(supp)q(\()2725 144 y(^)2715 171 y Fv(\030)t Fx(\),)195 511 y Fv(')266 446 y Ft(^)259 464 y FB(\030)259 538 y(N)326 511 y Fx(\()p Fv(x)p Fx(\))56 b(=)f Fv(')708 470 y Ft(\(1\))708 536 y FB(c)802 511 y Fx(\()p Fv(S)900 526 y FB(x)954 485 y Fx(^)944 511 y Fv(\030)5 b Fx(\))27 b Fv(;)212 b Fm(E)1370 557 y Fg(P)1431 511 y Fp(^)1425 524 y Fr(\030)1425 580 y(N)r(;x)1537 430 y Fo(\000)1600 511 y Fv(\037)p Fx(\()p Fv(\034)1759 446 y Ft(^)1752 464 y FB(\030)1741 538 y(N)1836 511 y Fv(>)27 b(t)p Fx(\))17 b Fv(')2100 446 y Ft(^)2093 464 y FB(\030)2093 538 y(N)2160 511 y Fx(\()p Fv(!)2259 526 y FB(t)2289 511 y Fx(\))2327 430 y Fo(\001)2428 511 y Fx(=)55 b Fm(E)2643 523 y Ft(^)2633 540 y Fg(P)2688 517 y Fr(c)2688 577 y(S)2726 585 y(x)2771 564 y Fp(^)2765 577 y Fr(\030)2803 430 y Fo(\000)2865 511 y Fv(\037)p Fx(\()p Fv(\034)3006 526 y FB(N)s(;x)3157 511 y Fv(>)27 b(t)p Fx(\))17 b Fv(')3414 470 y Ft(\(1\))3414 536 y FB(c)3508 511 y Fx(\()3556 485 y(^)3546 511 y Fv(\030)3589 526 y FB(t)3618 511 y Fx(\))3673 430 y Fo(\001)3746 511 y Fv(:)-24 767 y Fx(Therefore)334 1008 y Fm(E)424 1020 y Ft(^)408 1036 y Fq(P)463 1017 y Fr(c)499 927 y Fo(\000)561 1008 y Fv(A)641 943 y Ft(^)634 960 y FB(\030)634 1035 y Ft(3)p FB(;N)773 927 y Fo(\001)874 1008 y Fx(=)55 b Fm(E)1095 1020 y Ft(^)1079 1036 y Fq(P)1134 1017 y Fr(c)1170 897 y Fo(n)1236 1008 y Fv(m)1328 943 y Ft(^)1321 960 y FB(\030)1321 1035 y(N)1389 897 y Fo(h)1436 1008 y Fv(\037)p Fx(\()p Fv(x)28 b Fw(2)g Fv(C)1782 1023 y FB(M)1861 1008 y Fx(\))17 b Fv(')1980 967 y Ft(\(1\))1980 1032 y FB(c)2074 1008 y Fx(\()p Fv(S)2172 1023 y FB(x)2226 981 y Fx(^)2215 1008 y Fv(\030)5 b Fx(\))17 b Fm(E)2402 1020 y Ft(^)2392 1036 y Fg(P)2447 1014 y Fr(c)2447 1073 y(S)2485 1081 y(x)2529 1060 y Fp(^)2523 1073 y Fr(\030)2561 927 y Fo(\000)2623 1008 y Fv(\037)p Fx(\()p Fv(\034)2764 1023 y FB(N)s(;x)2915 1008 y Fv(>)28 b(t)p Fx(\))17 b Fv(')3173 967 y Ft(\(1\))3173 1032 y FB(c)3267 1008 y Fx(\()3315 981 y(^)3305 1008 y Fv(\030)3348 1023 y FB(t)3377 1008 y Fx(\))3432 927 y Fo(\001)3477 897 y(io)3607 1008 y Fv(:)-24 1220 y Fx(On)35 b(the)g(other)g(hand,)g(b)m(y)h(applying)f(the)g(Cauc)m(h)m(y-Sc)m(h)m (w)m(arz)j(inequalit)m(y)f(as)e(in)f(\(79\))g(and)h(due)g(to)f(Lemma) -24 1340 y(7,)f(w)m(e)g(obtain)414 1582 y(lim)398 1645 y FB(N)7 b Fq("1)583 1582 y Fm(E)673 1594 y Ft(^)657 1611 y Fq(P)712 1592 y Fr(c)748 1472 y Fo(n)814 1582 y Fv(m)906 1517 y Ft(^)899 1535 y FB(\030)899 1609 y(N)967 1472 y Fo(h)1014 1582 y Fv(\037)p Fx(\()p Fv(x)28 b Fw(2)g Fv(C)1360 1597 y FB(M)1439 1582 y Fx(\))17 b Fw(j)p Fv(')1586 1541 y Ft(\(1\))1586 1607 y FB(c)1679 1582 y Fx(\()p Fv(S)1777 1597 y FB(x)1831 1556 y Fx(^)1821 1582 y Fv(\030)t Fx(\))p Fw(j)g Fm(E)2035 1594 y Ft(^)2025 1611 y Fg(P)2080 1588 y Fr(c)2080 1648 y(S)2118 1656 y(x)2162 1635 y Fp(^)2156 1648 y Fr(\030)2194 1501 y Fo(\000)2257 1582 y Fv(\037)p Fx(\()p Fv(\034)2398 1597 y FB(N)s(;x)2548 1582 y Fw(\024)29 b Fv(t)p Fx(\))17 b Fw(j)p Fv(')2836 1541 y Ft(\(1\))2836 1607 y FB(c)2929 1582 y Fx(\()2977 1556 y(^)2967 1582 y Fv(\030)3010 1597 y FB(t)3039 1582 y Fx(\))p Fw(j)3122 1501 y Fo(\001)3167 1472 y(io)3336 1582 y Fx(=)55 b(0)28 b Fv(:)-24 1845 y Fx(The)34 b(last)f(t)m(w)m(o)g(iden)m(tities)i(imply) 411 2087 y(lim)394 2149 y FB(N)7 b Fq("1)580 2087 y Fm(E)670 2099 y Ft(^)654 2115 y Fq(P)709 2096 y Fr(c)744 2006 y Fo(\000)806 2087 y Fv(A)886 2022 y Ft(^)879 2040 y FB(\030)879 2114 y Ft(3)p FB(;N)1018 2006 y Fo(\001)1119 2087 y Fx(=)72 b(lim)1250 2149 y FB(N)7 b Fq("1)1436 2087 y Fm(E)1526 2099 y Ft(^)1510 2115 y Fq(P)1565 2096 y Fr(c)1601 1976 y Fo(n)1667 2087 y Fv(m)1759 2022 y Ft(^)1752 2040 y FB(\030)1752 2114 y(N)1820 1976 y Fo(h)1867 2087 y Fv(\037)p Fx(\()p Fv(x)28 b Fw(2)g Fv(C)2213 2102 y FB(M)2292 2087 y Fx(\))17 b Fv(')2411 2046 y Ft(\(1\))2411 2111 y FB(c)2504 2087 y Fx(\()p Fv(S)2602 2102 y FB(x)2656 2060 y Fx(^)2646 2087 y Fv(\030)5 b Fx(\))17 b Fm(E)2833 2099 y Ft(^)2823 2116 y Fg(P)2878 2093 y Fr(c)2878 2152 y(S)2916 2160 y(x)2960 2139 y Fp(^)2954 2152 y Fr(\030)2992 2006 y Fo(\000)3054 2087 y Fv(')3118 2046 y Ft(\(1\))3118 2111 y FB(c)3213 2087 y Fx(\()3261 2060 y(^)3251 2087 y Fv(\030)3294 2102 y FB(t)3323 2087 y Fx(\))3361 2006 y Fo(\001)3406 1976 y(io)3547 2087 y Fv(:)245 b Fx(\(81\))-24 2355 y(Observ)m(e)34 b(no)m(w)e(that)g(\(72\))f(remains)i(v)-5 b(alid)32 b(if)f(the)h(in)m(tegral)g(is)h(p)s(erformed)f(on)f Fv(C)2942 2370 y FB(M)3053 2355 y Fx(in)h(place)g(of)g Fv(C)3595 2370 y Ft(2)p FB(N)7 b Fq(\000)p Ft(2)3819 2355 y Fx(\(the)-24 2475 y(argumen)m(ts)34 b(used)g(in)f(the)g(pro)s (of)e(there)j(w)m(ork)f(also)g(in)g(this)g(case\))g(and)g(the)g (function)g Fv(h)p Fx(\()3286 2449 y(^)3276 2475 y Fv(\030)t Fx(\))g(is)g(de\014ned)h(as)836 2695 y Fv(h)p Fx(\()940 2669 y(^)930 2695 y Fv(\030)t Fx(\))56 b(=)f Fv(')1266 2654 y Ft(\(1\))1266 2720 y FB(c)1360 2695 y Fx(\()1408 2669 y(^)1398 2695 y Fv(\030)5 b Fx(\))17 b Fm(E)1585 2707 y Ft(^)1575 2724 y Fg(P)1630 2701 y Fr(c)1636 2748 y Fp(^)1630 2761 y Fr(\030)1667 2615 y Fo(\000)1730 2695 y Fv(')1794 2654 y Ft(\(1\))1794 2720 y FB(c)1888 2695 y Fx(\()1936 2669 y(^)1926 2695 y Fv(\030)1969 2710 y FB(t)1998 2695 y Fx(\))2036 2615 y Fo(\001)2137 2695 y Fx(=)55 b Fv(')2332 2654 y Ft(\(1\))2332 2720 y FB(c)2426 2695 y Fx(\()2474 2669 y(^)2464 2695 y Fv(\030)5 b Fx(\))2550 2615 y Fo(\000)2595 2695 y Fv(e)2640 2654 y FB(t)p Fq(L)2714 2662 y Fr(c)2751 2695 y Fv(')2815 2654 y Ft(\(1\))2815 2720 y FB(c)2909 2615 y Fo(\001)2955 2695 y Fx(\()3003 2669 y(^)2993 2695 y Fv(\030)t Fx(\))27 b Fv(:)-24 2973 y Fx(Note)47 b(that)g Fv(h)52 b Fw(2)g Fv(L)744 2937 y Ft(2)784 2973 y Fx(\()854 2948 y(^)822 2973 y Fw(N)904 2988 y Ft(0)943 2973 y Fv(;)1009 2948 y Fx(^)987 2973 y Fw(P)1064 2937 y FB(c)1056 2998 y Ft(0)1099 2973 y Fx(\).)85 b(Therefore)48 b(w)m(e)g(can)f(conclude)h(that)f(the)g (r.h.s.)86 b(of)47 b(\(81\))f(is)h(equal)g(to)-24 3107 y Fw(h)p Fv(')79 3056 y Ft(\(1\))79 3117 y FB(c)190 3107 y Fv(;)33 b(e)295 3071 y FB(t)p Fq(L)369 3079 y Fr(c)422 3107 y Fv(')486 3056 y Ft(\(1\))486 3117 y FB(c)580 3107 y Fw(i)635 3119 y Ft(^)619 3136 y Fq(P)674 3113 y Fr(c)668 3157 y Fp(0)710 3107 y Fx(.)3181 b Ff(2)-24 3376 y Fm(Lemma)39 b(7)49 b Fs(L)-5 b(et)36 b Fv(\034)707 3311 y Ft(^)700 3328 y FB(\030)689 3403 y(N)791 3376 y Fs(and)e Fv(\034)1022 3391 y FB(N)s(;x)1180 3376 y Fs(b)-5 b(e)35 b(de\014ne)-5 b(d)33 b(as)i(in)g Fx(\(77\))f Fs(and)g Fx(\(80\))p Fs(,)g(and)h(let)g Fv(M)j Fx(=)28 b(2)p Fv(N)k Fw(\000)23 b Fx(2[)p Fv(N)3494 3339 y FB(\013)3544 3376 y Fx(])p Fs(.)44 b(Then)911 3716 y Fx(lim)894 3779 y FB(N)7 b Fq("1)1107 3716 y Fm(E)1197 3728 y Ft(^)1181 3744 y Fq(P)1236 3726 y Fr(c)1272 3605 y Fo(n)1338 3716 y Fv(m)1430 3651 y Ft(^)1423 3669 y FB(\030)1423 3743 y(N)1491 3605 y Fo(h)1538 3716 y Fv(\037)p Fx(\()p Fv(x)28 b Fw(2)g Fv(C)1884 3731 y FB(M)1963 3716 y Fx(\))17 b Fm(P)2102 3651 y Ft(^)2095 3669 y FB(\030)2095 3743 y(N)s(;x)2217 3635 y Fo(\000)2279 3716 y Fv(\034)2339 3651 y Ft(^)2332 3669 y FB(\030)2321 3743 y(N)2416 3716 y Fw(\024)28 b Fv(t)2573 3635 y Fo(\001)2619 3605 y(io)2788 3716 y Fx(=)55 b(0)27 b Fv(;)797 b Fx(\(82\))911 3949 y(lim)894 4012 y FB(N)7 b Fq("1)1107 3949 y Fm(E)1197 3961 y Ft(^)1181 3978 y Fq(P)1236 3959 y Fr(c)1272 3839 y Fo(n)1338 3949 y Fv(m)1430 3885 y Ft(^)1423 3902 y FB(\030)1423 3976 y(N)1491 3839 y Fo(h)1538 3949 y Fv(\037)p Fx(\()p Fv(x)28 b Fw(2)g Fv(C)1884 3964 y FB(M)1963 3949 y Fx(\))2048 3924 y(^)2034 3949 y Fm(P)2111 3908 y FB(c)2111 3988 y(S)2154 3996 y Fr(x)2199 3970 y Ft(^)2192 3988 y FB(\030)2230 3869 y Fo(\000)2292 3949 y Fv(\034)2334 3964 y FB(N)s(;x)2485 3949 y Fw(\024)g Fv(t)2642 3869 y Fo(\001)2688 3839 y(io)2856 3949 y Fx(=)56 b(0)27 b Fv(:)728 b Fx(\(83\))-24 4218 y Fm(Pro)s(of.)82 b Fx(One)42 b(can)g(c)m(hec)m(k)i(b)m(y)f(a)f(coupling)g(argumen)m(t)h(that)e(the)i (t)m(w)m(o)f(exp)s(ectations)i(in)e(\(82\))f(and)h(\(83\))-24 4338 y(coincide:)i(for)29 b(eac)m(h)i Fv(N)38 b Fw(2)28 b Fu(N)1032 4353 y Ft(+)1091 4338 y Fx(,)1159 4312 y(^)1149 4338 y Fv(\030)k Fw(2)1350 4313 y Fx(^)1318 4338 y Fw(N)44 b Fx(and)30 b Fv(x)e Fw(2)g Fv(C)1878 4353 y FB(M)1974 4338 y Fw(\\)17 b Fx(supp)q(\()2306 4312 y(^)2296 4338 y Fv(\030)t Fx(\),)31 b(one)f(can)g(de\014ne)h(a)f(probabilit)m(y)h (measure)-24 4482 y Fv(\026)i Fx(on)f(\012)280 4417 y Ft(^)273 4435 y FB(\030)273 4509 y(N)363 4482 y Fw(\002)471 4457 y Fx(^)463 4482 y(\004)g(suc)m(h)i(that)131 4728 y Fv(\026)p Fx(\()p Fv(A)22 b Fw(\002)431 4703 y Fx(^)423 4728 y(\004\))55 b(=)g Fm(P)796 4663 y Ft(^)789 4681 y FB(\030)789 4755 y(N)s(;x)911 4728 y Fx(\()p Fv(A)p Fx(\))28 b Fv(;)183 b(\026)p Fx(\(\012)1472 4663 y Ft(^)1465 4681 y FB(\030)1465 4755 y(N)1554 4728 y Fw(\002)23 b Fv(B)5 b Fx(\))55 b(=)1971 4702 y(^)1957 4728 y Fm(P)2034 4687 y FB(c)2034 4766 y(S)2077 4774 y Fr(x)2123 4749 y Ft(^)2116 4766 y FB(\030)2154 4728 y Fx(\()p Fv(B)5 b Fx(\))27 b Fv(;)212 b Fw(8)28 b Fv(A)g Fw(2)g(B)s Fx(\(\012)3036 4663 y Ft(^)3029 4681 y FB(\030)3029 4755 y(N)3097 4728 y Fx(\))g Fv(;)99 b Fw(8)28 b Fv(B)33 b Fw(2)28 b(B)s Fx(\()3687 4703 y(^)3679 4728 y(\004)q(\))f Fv(;)-24 4998 y Fx(and)43 b(suc)m(h)h(that,)g Fv(\026)e Fx(almost)h(surely)-8 b(,)46 b Fv(\034)1459 4933 y Ft(^)1452 4951 y FB(\030)1440 5025 y(N)1508 4998 y Fx(\()p Fv(!)p 1546 5014 65 4 v 3 w Fx(\))e(=)h Fv(\034)1855 5013 y FB(N)s(;x)1978 4998 y Fx(\()2026 4972 y(^)2016 4998 y Fv(\030)p 2016 5033 48 4 v 4 w Fx(\))d(and)h Fv(!)2404 5013 y FB(s)2485 4998 y Fx(=)h Fv(x)29 b Fx(+)g Fv(X)2875 5013 y FB(s)2911 4998 y Fx(\()2959 4972 y(^)2949 4998 y Fv(\030)p 2949 5033 V 5 w Fx(\))42 b(for)g(an)m(y)h(0)h Fw(\024)h Fv(s)f(<)g(\034)3916 4933 y Ft(^)3909 4951 y FB(\030)3897 5025 y(N)3965 4998 y Fx(.)-24 5156 y(Suc)m(h)35 b(a)e(coupling)g Fv(\026)g Fx(implies)1123 5131 y(^)1109 5156 y Fm(P)1186 5120 y FB(c)1186 5200 y(S)1229 5208 y Fr(x)1274 5182 y Ft(^)1267 5200 y FB(\030)1305 5076 y Fo(\000)1367 5156 y Fv(\034)1409 5171 y FB(N)s(;x)1561 5156 y Fw(\024)c Fv(t)1719 5076 y Fo(\001)1793 5156 y Fx(=)g Fm(P)1982 5092 y Ft(^)1975 5109 y FB(\030)1975 5183 y(N)s(;x)2097 5076 y Fo(\000)2159 5156 y Fv(\034)2220 5092 y Ft(^)2213 5109 y FB(\030)2201 5183 y(N)2298 5156 y Fw(\024)g Fv(t)2456 5076 y Fo(\001)2501 5156 y Fx(.)46 b(Th)m(us)34 b(w)m(e)g(need)h(to)e(pro)m(v)m(e)h(only)g (\(82\).)-24 5290 y(Moreo)m(v)m(er,)h(without)e(loss)g(of)f(generalit)m (y)-8 b(,)34 b(w)m(e)g(assume)g Fv(r)2085 5305 y FB(c)2147 5290 y Fx(=)28 b(1.)1935 5564 y(34)p eop end %%Page: 35 35 TeXDict begin 35 34 bop 123 171 a Fx(T)-8 b(o)22 b(this)g(aim)g(let)g (us)g(co)m(v)m(er)h Fv(C)1169 186 y Ft(2)p FB(N)7 b Fq(\000)p Ft(2)1362 171 y Fw(n)p Fv(C)1482 186 y FB(M)1582 171 y Fx(b)m(y)23 b(disjoin)m(t)g(cub)s(es)g Fv(C)2372 186 y Ft(1)p FB(;i)2476 171 y Fx(of)e(side)i(1,)h Fv(i)k Fw(2)g Fv(I)8 b 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Fw(n)12 b Fv(C)2343 1459 y FB(M)2449 1444 y Fx(and)28 b Fv(t)2669 1459 y Ft(1)2736 1444 y Fx(b)s(e)g(the)g(time)h(sp)s(en)m(t)f(in)g Fv(x)3664 1459 y Ft(1)3732 1444 y Fx(b)s(efore)-24 1565 y(jumping)34 b(a)m(w)m(a)m(y)-8 b(.)47 b(The)34 b(index)h Fv(i)1133 1580 y Ft(1)1206 1565 y Fx(is)f(c)m(haracterized)h(b)m(y)f (the)g(requiremen)m(t)i(that)d Fv(x)3014 1580 y Ft(1)3083 1565 y Fw(2)c Fv(C)3248 1580 y Ft(1)p FB(;i)3327 1589 y Fp(1)3365 1565 y Fx(.)46 b(Supp)s(ose)34 b(no)m(w)-24 1685 y(that)c Fv(i)218 1700 y Ft(1)258 1685 y Fv(;)17 b(:)g(:)g(:)e(;)i(i)509 1700 y FB(j)546 1685 y Fx(,)30 b Fv(x)658 1700 y Ft(1)698 1685 y Fv(;)17 b(:)g(:)g(:)f(;)h(x)972 1700 y FB(j)1038 1685 y Fx(and)30 b Fv(t)1260 1700 y Ft(1)1300 1685 y Fv(;)17 b(:)g(:)g(:)f(;)h(t)1554 1700 y FB(j)1620 1685 y Fx(ha)m(v)m(e)31 b(b)s(een)g(de\014ned)g(and)f(that) f Fv(j)34 b(<)28 b(k)s Fx(.)42 b(Then)31 b Fv(x)3406 1700 y FB(j)t Ft(+1)3563 1685 y Fx(is)f(the)h(\014rst)-24 1832 y(p)s(oin)m(t)k(in)f Fv(C)418 1847 y Ft(2)p FB(N)7 b Fq(\000)p Ft(2)634 1832 y Fw(n)707 1751 y Fo(\000)753 1832 y Fv(C)823 1847 y FB(M)925 1832 y Fw([)23 b Fv(C)1084 1847 y Ft(1)p FB(;i)1163 1856 y Fp(1)1225 1832 y Fw([)h(\001)17 b(\001)g(\001)k([)j Fv(C)1614 1847 y Ft(1)p FB(;i)1693 1857 y Fr(j)1729 1751 y Fo(\001)1809 1832 y Fx(visited)36 b(during)e(the)h(time)g(in)m(terv)-5 b(al)35 b([0)p Fv(;)17 b(\034)3361 1767 y Ft(^)3354 1785 y FB(\030)3343 1859 y(N)3410 1832 y Fx(\()p Fv(!)p 3448 1848 V 3 w Fx(\))g(\))34 b(and)g Fv(t)3865 1847 y FB(j)t Ft(+1)-24 1953 y Fx(is)f(the)g(time)h (sp)s(en)m(t)f(at)g Fv(x)900 1968 y FB(j)t Ft(+1)1059 1953 y Fx(during)g(suc)m(h)h(a)e(\014rst)i(visit.)44 b(Moreo)m(v)m(er,)34 b Fv(i)2611 1968 y FB(j)t Ft(+1)2771 1953 y Fx(is)f(suc)m(h)h(that)e Fv(x)3355 1968 y FB(j)t Ft(+1)3510 1953 y Fw(2)c Fv(C)3674 1968 y Ft(1)p FB(;i)3753 1978 y Fr(j)s Fp(+1)3867 1953 y Fx(.)123 2154 y(No)m(w)k(let)f Fv(T)561 2089 y Ft(^)554 2107 y FB(\030)540 2179 y(i)592 2154 y Fx(,)g Fv(i)d Fw(2)g Fv(I)40 b Fx(and)1086 2127 y(^)1076 2154 y Fv(\030)32 b Fw(2)1277 2129 y Fx(^)1245 2154 y Fw(N)15 b Fx(,)31 b(b)s(e)h(a)f(family)h(of)e(indep)s(enden)m(t) k(exp)s(onen)m(tial)f(random)e(v)-5 b(ariables)32 b(\(all)-24 2298 y(indep)s(enden)m(t)j(from)d(the)h(ab)s(o)m(v)m(e)h(random)f(ob)5 b(jects\))33 b(and)g(suc)m(h)h(that)e Fv(T)2632 2234 y Ft(^)2625 2251 y FB(\030)2611 2324 y(i)2695 2298 y Fx(has)h(parameter)3346 2272 y(^)3335 2298 y Fv(\030)3383 2218 y Fo(\000)3464 2273 y(e)3445 2298 y Fv(C)3515 2313 y Ft(1)p FB(;i)3615 2298 y Fx(\),)f(where)1230 2513 y Fo(e)1211 2538 y Fv(C)1281 2553 y Ft(1)p FB(;i)1419 2538 y Fx(=)55 b Fw(f)p Fv(y)31 b Fw(2)d Fu(R)1845 2497 y FB(d)1930 2538 y Fx(:)44 b(dist)q(\()p Fv(y)t(;)17 b(C)2363 2553 y Ft(1)p FB(;i)2444 2538 y Fx(\))28 b Fw(\024)g Fx(1)17 b Fw(g)p Fv(:)-24 2770 y Fx(Since,)35 b(giv)m(en)525 2743 y(^)515 2770 y Fv(\030)t Fx(,)f Fv(k)i Fx(and)e(\()p Fv(x)994 2785 y Ft(1)1034 2770 y Fv(;)17 b(:)g(:)g(:)f(;)h(x)1308 2785 y FB(k)1350 2770 y Fx(\),)34 b Fv(t)1484 2785 y FB(j)1554 2770 y Fx(\(1)29 b Fw(\024)h Fv(j)35 b Fw(\024)29 b Fv(k)s Fx(\))34 b(are)f(indep)s(enden)m(t)j(exp)s(onen)m(tial)f(v)-5 b(ariables)34 b(and)f Fv(t)3955 2785 y FB(j)-24 2903 y Fx(has)g(parameter)g(non)g(larger)f(than)1320 2877 y(^)1310 2903 y Fv(\030)1358 2822 y Fo(\000)1439 2878 y(e)1420 2903 y Fv(C)1490 2918 y Ft(1)p FB(;i)1569 2928 y Fr(j)1622 2822 y Fo(\001)1700 2903 y Fx(and)h(since)h Fv(k)c Fw(\025)f Fv(k)2367 2918 y Ft(min)2516 2903 y Fx(:=)e([)p Fv(N)2761 2867 y FB(\013)2811 2903 y Fx(])c Fw(\000)f Fx(1,)33 b(w)m(e)g(obtain)54 3160 y Fm(E)144 3172 y Ft(^)128 3189 y Fq(P)183 3170 y Fr(c)218 3050 y Fo(n)285 3160 y Fv(m)377 3096 y Ft(^)370 3113 y FB(\030)370 3187 y(N)437 3050 y Fo(h)484 3160 y Fv(\037)p Fx(\()p Fv(x)28 b Fw(2)g Fv(C)830 3175 y FB(M)909 3160 y Fx(\))17 b Fm(P)1048 3096 y Ft(^)1041 3113 y FB(\030)1041 3187 y(N)s(;x)1163 3080 y Fo(\000)1225 3160 y Fv(\034)1286 3096 y Ft(^)1279 3113 y FB(\030)1267 3187 y(N)1363 3160 y Fw(\024)28 b Fv(t)1520 3080 y Fo(\001)1565 3050 y(io)181 3450 y Fx(=)394 3318 y Fq(j)p FB(I)5 b Fq(j)360 3356 y Fo(X)312 3568 y FB(n)p Ft(=)p FB(k)447 3579 y Fp(min)570 3356 y Fo(X)568 3568 y FB(l)p 568 3580 22 3 v 1 w Fq(2)p FB(I)673 3549 y Fr(n)668 3584 y Fk(\003)869 3356 y Fo(X)732 3572 y FB(y)p 732 3597 38 3 v 2 w Fq(2)816 3523 y Fi(Q)883 3543 y Fr(n)883 3593 y(j)s Fp(=1)993 3572 y FB(C)1043 3584 y Fp(1)p Fr(;l)1112 3600 y(j)1166 3450 y Fm(E)1256 3462 y Ft(^)1240 3479 y Fq(P)1295 3460 y Fr(c)1330 3340 y Fo(n)1397 3450 y Fv(m)1489 3386 y Ft(^)1482 3403 y FB(\030)1482 3477 y(N)1549 3340 y Fo(h)1596 3450 y Fv(\037)p Fx(\()p Fv(x)28 b Fw(2)g Fv(C)1942 3465 y FB(M)2021 3450 y Fx(\))17 b Fm(P)2160 3386 y Ft(^)2153 3403 y FB(\030)2153 3477 y(N)s(;x)2275 3370 y Fo(\000)2337 3450 y Fv(\034)2398 3386 y Ft(^)2391 3403 y FB(\030)2379 3477 y(N)2475 3450 y Fw(\024)28 b Fv(t;)33 b(k)e Fx(=)d Fv(n;)17 b(x)3018 3465 y Ft(1)3085 3450 y Fx(=)28 b Fv(y)3237 3465 y Ft(1)3276 3450 y Fv(;)17 b(:)g(:)g(:)e(;)i(x)3549 3465 y FB(n)3624 3450 y Fx(=)28 b Fv(y)3776 3465 y FB(n)3822 3370 y Fo(\001)3868 3340 y(io)181 3845 y Fw(\024)396 3713 y Fq(j)p FB(I)5 b Fq(j)362 3750 y Fo(X)314 3962 y FB(n)p Ft(=)p FB(k)449 3973 y Fp(min)572 3750 y Fo(X)570 3962 y FB(l)p 570 3974 22 3 v 1 w Fq(2)p FB(I)675 3944 y Fr(n)670 3979 y Fk(\003)734 3845 y Fm(E)824 3857 y Ft(^)808 3874 y Fq(P)863 3855 y Fr(c)898 3734 y Fo(n)965 3845 y Fv(m)1057 3780 y Ft(^)1050 3798 y FB(\030)1050 3872 y(N)1117 3734 y Fo(h)1164 3845 y Fv(\037)p Fx(\()p Fv(x)29 b Fw(2)f Fv(C)1511 3860 y FB(M)1589 3845 y Fx(\))17 b Fm(P)1728 3780 y Ft(^)1721 3798 y FB(\030)1721 3872 y(N)s(;x)1843 3764 y Fo(\000)1889 3845 y Fv(k)31 b Fx(=)c Fv(n;)17 b(i)2209 3860 y Ft(1)2276 3845 y Fx(=)28 b Fv(l)2409 3860 y Ft(1)2448 3845 y Fv(;)17 b(:)g(:)g(:)f(;)h(i)2700 3860 y FB(n)2775 3845 y Fx(=)27 b Fv(l)2907 3860 y FB(n)2987 3845 y Fx(\))3025 3734 y Fo(i)3094 3845 y Fw(\001)697 b Fx(\(84\))2722 4137 y(Prob)2962 4056 y Fo(\000)3007 4137 y Fv(T)3085 4072 y Ft(^)3078 4090 y FB(\030)3064 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4619 y Fw(\024)28 b Fv(t)p Fx(\))17 b Fv(:)123 4819 y Fx(Let)35 b(us)h(de\014ne)g Fv(m)c Fx(:=)g Fm(E)1054 4831 y Ft(^)1038 4847 y Fq(P)1093 4828 y Fr(c)1128 4738 y Fo(\000)1201 4793 y Fx(^)1191 4819 y Fv(\030)t Fx(\()1295 4794 y Fo(e)1276 4819 y Fv(C)1346 4834 y Ft(1)1385 4819 y Fx(\))1440 4738 y Fo(\001)1485 4819 y Fx(,)k(where)1851 4794 y Fo(e)1832 4819 y Fv(C)1902 4834 y Ft(1)1973 4819 y Fx(=)c Fw(f)p Fv(y)j Fw(2)d Fu(R)2384 4783 y FB(d)2456 4819 y Fx(:)g(dist)q(\()p Fv(y)t(;)17 b(C)2877 4834 y Ft(1)2915 4819 y Fx(\))32 b Fw(\024)g Fx(1)p Fw(g)p Fx(.)51 b(Giv)m(en)36 b Fv(\024)c(>)f Fx(0)k(and)-24 4939 y Fv(l)p -24 4955 32 4 v 30 w Fw(2)28 b Fv(I)180 4903 y FB(n)172 4964 y Fq(\003)260 4939 y Fx(as)33 b(ab)s(o)m(v)m(e,)g(w)m(e)h(de\014ne)f Fw(A)28 b Fx(=)f Fw(A)p Fx(\()p Fv(\024;)17 b(l)p 1537 4955 V 2 w Fx(\))32 b(as)h(follo)m(ws)637 5180 y Fw(A)55 b Fx(=)903 5069 y Fo(n)996 5154 y Fx(^)986 5180 y Fv(\030)32 b Fw(2)1187 5155 y Fx(^)1155 5180 y Fw(N)59 b Fx(:)1384 5095 y Fo(\014)1384 5155 y(\014)1433 5099 y(\010)1492 5180 y Fv(j)50 b Fx(:)44 b(1)28 b Fw(\024)g Fv(j)33 b Fw(\024)c Fv(n)j Fx(and)2304 5154 y(^)2294 5180 y Fv(\030)2341 5099 y Fo(\000)2406 5155 y(e)2387 5180 y Fv(C)2457 5195 y Ft(1)p FB(;l)2533 5205 y Fr(j)2569 5099 y Fo(\001)2643 5180 y Fv(>)27 b(\024)17 b(m)2921 5099 y Fo(\011)2979 5095 y(\014)2979 5155 y(\014)3040 5180 y Fv(>)3154 5113 y(n)p 3154 5157 59 4 v 3159 5248 a Fx(2)3238 5069 y Fo(o)3305 5180 y Fv(:)1935 5564 y Fx(35)p eop end %%Page: 36 36 TeXDict begin 36 35 bop -24 171 a Fx(Then,)34 b(b)m(y)g(the)f(Cheb)m (yshev)j(inequalit)m(y)e(and)f(the)g(stationarit)m(y)g(of)2523 145 y(^)2500 171 y Fw(P)2577 134 y FB(c)2612 171 y Fx(,)89 392 y(^)67 417 y Fw(P)144 376 y FB(c)179 336 y Fo(\000)241 417 y Fw(A)338 336 y Fo(\001)438 417 y Fw(\024)586 350 y Fx(2)p 581 394 59 4 v 581 485 a Fv(n)677 417 y Fm(E)767 429 y Ft(^)751 446 y Fq(P)806 427 y Fr(c)842 306 y Fo(\020)901 332 y(\014)901 392 y(\014)951 336 y(\010)1009 417 y Fv(j)50 b Fx(:)45 b(1)27 b Fw(\024)h Fv(j)34 b Fw(\024)28 b Fv(n)33 b Fx(and)1822 391 y(^)1812 417 y Fv(\030)1859 336 y Fo(\000)1924 392 y(e)1905 417 y Fv(C)1975 432 y Ft(1)p FB(;l)2051 442 y Fr(j)2087 336 y Fo(\001)2160 417 y Fv(>)28 b(\024)17 b(m)2439 336 y Fo(\011)2497 332 y(\014)2497 392 y(\014)2530 336 y(\001)2631 417 y Fw(\024)56 b Fx(2)2852 392 y(^)2830 417 y Fw(P)2907 376 y FB(c)2941 336 y Fo(\000)3014 391 y Fx(^)3004 417 y Fv(\030)t Fx(\()3108 392 y Fo(e)3089 417 y Fv(C)3159 432 y Ft(1)3198 417 y Fx(\))28 b Fv(>)f(\024)17 b(m)3542 336 y Fo(\001)3643 417 y Fw(!)55 b Fx(0)28 b Fv(;)-24 650 y Fx(as)33 b Fv(\024)28 b Fw(!)f(1)p Fx(.)43 b(Note)33 b(that)f(the)h(complemen)m(t)i Fw(A)1717 614 y FB(c)1784 650 y Fx(of)d Fw(A)g Fx(can)h(b)s(e)g(written)g(as)526 879 y Fw(A)606 838 y FB(c)696 879 y Fx(=)827 769 y Fo(n)920 853 y Fx(^)910 879 y Fv(\030)f Fw(2)1111 854 y Fx(^)1079 879 y Fw(N)59 b Fx(:)1308 795 y Fo(\014)1308 854 y(\014)1358 799 y(\010)1416 879 y Fv(j)50 b Fx(:)44 b(1)28 b Fw(\024)g Fv(j)33 b Fw(\024)c Fv(n)j Fx(and)2228 853 y(^)2218 879 y Fv(\030)2265 799 y Fo(\000)2330 854 y(e)2311 879 y Fv(C)2381 894 y Ft(1)p FB(;l)2457 904 y Fr(j)2494 799 y Fo(\001)2567 879 y Fw(\024)c Fv(\024)17 b(m)2847 799 y Fo(\011)2905 795 y(\014)2905 854 y(\014)2966 879 y Fw(\025)3071 799 y Fo(\002)3139 812 y Fv(n)p 3139 856 V 3144 948 a Fx(2)3224 799 y Fo(\003)3265 918 y Fq(\003)3321 769 y Fo(o)3415 879 y Fv(;)-24 1139 y Fx(where)38 b([)p Fv(n=)p Fx(2])472 1154 y Fq(\003)548 1139 y Fx(is)g(de\014ned)g(as)f Fv(n=)p Fx(2)f(for)g Fv(n)h Fx(ev)m(en)h(and)f(as)g([)p Fv(n=)p Fx(2])25 b(+)g(1)36 b(for)g Fv(n)h Fx(o)s(dd.)56 b(If)3123 1112 y(^)3113 1139 y Fv(\030)39 b Fw(2)c(A)3376 1102 y FB(c)3447 1139 y Fx(then)i(at)f(least)-24 1202 y Fo(\002)44 1243 y FB(n)p 44 1260 43 4 v 48 1317 a Ft(2)114 1202 y Fo(\003)155 1322 y Fq(\003)226 1283 y Fx(of)c(the)g(exp)s(onen)m (tial)h(v)-5 b(ariables)32 b Fv(T)1507 1218 y Ft(^)1500 1235 y FB(\030)1486 1310 y(l)1507 1319 y Fp(1)1546 1283 y Fx(,)g(.)16 b(.)g(.)g(,)p Fv(T)1840 1218 y Ft(^)1833 1235 y FB(\030)1818 1310 y(l)1839 1318 y Fr(n)1919 1283 y Fx(ha)m(v)m(e)33 b(parameter)f(non)g(larger)g(than)f Fv(\024)17 b(m)p Fx(.)44 b(Then,)33 b(b)m(y)f(a)-24 1426 y(coupling)i(argumen)m(t)f(\()p Fn(e.g.)43 b Fx(the)33 b(pro)s(of)f(of)g(Prop)s(osition)h(2\),)f(w)m(e)i(get)f(for)f(all)2836 1399 y(^)2826 1426 y Fv(\030)g Fw(2)c(A)3075 1389 y FB(c)624 1709 y Fx(Prob)831 1628 y Fo(\000)877 1709 y Fv(T)954 1644 y Ft(^)947 1662 y FB(\030)934 1737 y(l)955 1746 y Fp(1)1016 1709 y Fx(+)22 b Fw(\001)17 b(\001)g(\001)j Fx(+)i Fv(T)1428 1644 y Ft(^)1421 1662 y FB(\030)1407 1737 y(l)1428 1745 y Fr(n)1503 1709 y Fw(\024)28 b Fv(t)p Fx(\))55 b Fw(\024)h Fv(e)1914 1668 y Fq(\000)p FB(\024)12 b(mt)2234 1584 y Fq(1)2197 1614 y Fo(X)2130 1830 y FB(r)r Ft(=[)p FB(n=)p Ft(2])2372 1838 y Fk(\003)2434 1642 y Fx(\()p Fv(\024)17 b(m)g(t)p Fx(\))2720 1605 y FB(r)p 2434 1686 324 4 v 2559 1777 a Fv(r)s Fx(!)2823 1709 y(=:)56 b Fv(\036)p Fx(\()p Fv(\024;)33 b(n)p Fx(\))28 b Fv(:)-24 2027 y Fx(Due)33 b(to)f(the)h(ab)s(o)m(v)m(e)h(estimates)g(and)e(since) i Fv(n)28 b Fw(\025)h Fv(k)1845 2042 y Ft(min)1994 2027 y Fx(=)e([)p Fv(N)2212 1991 y FB(\013)2262 2027 y Fx(])c Fw(\000)f Fx(1,)33 b(w)m(e)g(get)886 2244 y(r.h.s.)44 b(of)32 b(\(84\))87 b Fw(\024)56 b Fx(2)1745 2219 y(^)1723 2244 y Fw(P)1800 2203 y FB(c)1835 2164 y Fo(\000)1907 2218 y Fx(^)1897 2244 y Fv(\030)t Fx(\()2001 2219 y Fo(e)1982 2244 y Fv(C)2052 2259 y Ft(1)2092 2244 y Fx(\))27 b Fv(>)h(\024)17 b(m)2436 2164 y Fo(\001)2531 2244 y Fx(+)50 b Fv(\036)p Fx(\()p Fv(\024;)17 b(N)2941 2203 y FB(\013)2990 2244 y Fx(\))28 b Fv(:)-24 2452 y Fx(The)34 b(lemma)f(follo)m(ws)h(b)m(y)f (taking)g(\014rst)g(the)g(limit)g Fv(N)38 b Fw(")28 b(1)k Fx(and)g(then)i(the)f(limit)g Fv(\024)27 b Fw(")h(1)p Fx(.)641 b Ff(2)-24 2739 y Fl(6.3)136 b(Random)45 b(resistor)h(net)l(w) l(orks)-24 2924 y Fx(W)-8 b(e)41 b(conclude)h(this)f(section)h(b)m(y)f (p)s(oin)m(ting)g(out)f(that)g(the)h(di\013usion)h(co)s(e\016cien)m(t)g Fv(D)3161 2859 y Ft(^)3154 2876 y FB(\030)3151 2951 y(N)3258 2924 y Fx(of)e(the)h(p)s(erio)s(dized)-24 3084 y(medium)g(can)f(b)s(e)g (expressed)i(in)e(terms)h(of)e(the)h(e\013ectiv)m(e)i(conductance)f(of) e(the)h(graph)g(\()p 3384 3004 70 4 v Fw(V)3460 3005 y Ft(^)3453 3023 y FB(\030)3453 3109 y(N)3520 3084 y Fv(;)p 3564 3004 62 4 v 17 w Fw(E)3632 3005 y Ft(^)3625 3023 y FB(\030)3625 3109 y(N)3692 3084 y Fx(\))g(when)-24 3205 y(assigning)33 b(suitable)g(b)s(ond)f(conductances.)46 b(More)32 b(precisely)-8 b(,)34 b(consider)g(the)e(electrical)h(net)m (w)m(ork)h(giv)m(en)f(b)m(y)-24 3358 y(the)f(graph)e(\()p 455 3278 70 4 v Fw(V)532 3278 y Ft(^)525 3296 y FB(\030)525 3382 y(N)592 3358 y Fv(;)p 636 3278 62 4 v 17 w Fw(E)704 3278 y Ft(^)697 3296 y FB(\030)697 3382 y(N)764 3358 y Fx(\))h(where)h(the)f(b)s(ond)g Fw(f)p Fv(x;)17 b(y)t Fw(g)27 b(2)p 1896 3278 V 28 w(E)1964 3278 y Ft(^)1957 3296 y FB(\030)1957 3382 y(N)2055 3358 y Fx(has)32 b(conductivit)m(y)h Fv(c)p Fx(\()p Fw(f)p Fv(\031)t Fx(\()p Fv(x)p Fx(\))p Fv(;)17 b(\031)t Fx(\()p Fv(y)t Fx(\))p Fw(g)p Fx(\))29 b(with)j Fv(c)p Fx(\()p Fw(f\001)p Fv(;)17 b Fw(\001g)p Fx(\))-24 3502 y(de\014ned)38 b(in)e(\(63\).)53 b(Then,)39 b(the)d(e\013ectiv)m(e)i(conductance)g Fv(G)2171 3437 y Ft(^)2164 3455 y FB(\030)2164 3529 y(N)2267 3502 y Fx(of)e(this)g(net)m(w)m(ork)i(is)f(de\014ned)g(as)f(the)h(curren)m(t) -24 3622 y(\015o)m(wing)e(from)g(\000)605 3581 y Fq(\000)605 3649 y FB(N)706 3622 y Fx(to)f(\000)888 3581 y Ft(+)888 3649 y FB(N)989 3622 y Fx(when)i(a)e(unit)h(p)s(oten)m(tial)f (di\013erence)i(b)s(et)m(w)m(een)h(\000)2828 3581 y Fq(\000)2828 3649 y FB(N)2929 3622 y Fx(to)d(\000)3111 3581 y Ft(+)3111 3649 y FB(N)3213 3622 y Fx(is)h(imp)s(osed.)49 b(It)35 b(can)-24 3743 y(b)s(e)40 b(calculated)h(from)e(Ohm's)i(la)m(w)f(and)g (the)g(Kirc)m(hho\013)g(rule)g(as)g(follo)m(ws.)66 b(Let)40 b(the)g(electrical)h(p)s(oten)m(tial)-24 3863 y Fv(V)22 b Fx(\()p Fv(x)p Fx(\))33 b(v)-5 b(anish)33 b(on)g(the)g(left)f(b)s (order)h(\000)1367 3822 y Fq(\000)1367 3890 y FB(N)1434 3863 y Fx(,)g(b)s(e)f(equal)i(to)e(1)g(on)h(the)g(righ)m(t)g(b)s(order) f(\000)2996 3822 y Ft(+)2996 3890 y FB(N)3063 3863 y Fx(,)h(and)g(satisfy:)658 4002 y Fo(X)532 4255 y FB(y)14 b Ft(:)d Fq(f)p FB(y)r(;x)p Fq(g2)p 826 4201 43 3 v(E)876 4202 y Fp(^)870 4215 y Fr(\030)870 4272 y(N)944 4096 y Fv(c)p Fx(\()p Fw(f)p Fv(\031)t Fx(\()p Fv(x)p Fx(\))p Fv(;)17 b(\031)t Fx(\()p Fv(y)t Fx(\))p Fw(g)p Fx(\))1609 4016 y Fo(\000)1655 4096 y Fv(V)k Fx(\()p Fv(y)t Fx(\))g Fw(\000)i Fv(V)e Fx(\()p Fv(x)p Fx(\))2191 4016 y Fo(\001)2293 4096 y Fx(=)55 b(0)143 b(for)32 b(an)m(y)144 b Fv(x)28 b Fw(2)g Fv(Q)3321 4032 y Ft(^)3314 4049 y FB(\030)3314 4123 y(N)3409 4096 y Fv(:)-24 4452 y Fx(Then)50 b(the)e(e\013ectiv)m(e) i(conductance)g(is)e(giv)m(en)h(b)m(y)g(the)g(curren)m(t)g(\015o)m (wing)g(through)f(the)g(surfaces)h Fw(f)p Fv(x)55 b Fw(2)-24 4572 y Fx([)p Fw(\000)p Fv(N)5 b(;)17 b(N)10 b Fx(])322 4536 y FB(d)392 4572 y Fx(:)27 b Fv(x)501 4536 y Ft(\(1\))624 4572 y Fx(=)g Fw(\006)p Fv(N)10 b Fw(g)p Fx(:)1130 4804 y Fv(G)1214 4739 y Ft(^)1207 4756 y FB(\030)1207 4831 y(N)1330 4804 y Fx(=)1502 4709 y Fo(X)1461 4955 y FB(x)p Fq(2)p FB(B)1610 4910 y Fp(^)1604 4923 y Fr(\030)r Fk(\000)1601 4978 y Fr(N)1703 4804 y Fv(V)21 b Fx(\()p Fv(x)p Fx(\))56 b(=)2139 4709 y Fo(X)2099 4955 y FB(x)p Fq(2)p FB(B)2248 4910 y Fp(^)2242 4923 y Fr(\030)r Fp(+)2239 4978 y Fr(N)2323 4723 y Fo(\000)2369 4804 y Fx(1)22 b Fw(\000)g Fv(V)g Fx(\()p Fv(x)p Fx(\))2749 4723 y Fo(\001)2811 4804 y Fv(:)981 b Fx(\(85\))-24 5195 y(By)36 b(a)e(w)m(ell-kno)m(wn)j(analogy) e(it)g(is)g(link)m(ed)i(to)d(the)h(di\013usion)h(co)s(e\016cien)m(t)h Fv(D)2813 5130 y Ft(^)2806 5147 y FB(\030)2803 5222 y(N)2905 5195 y Fx(\(see)f Fn(e.g.)50 b Fx([DF)m(GW)o(,)36 b(Prop)s(o-)-24 5315 y(sition)d(4.15])g(for)f(a)g(similar)h(pro)s(of)7 b(\):)1935 5564 y(36)p eop end %%Page: 37 37 TeXDict begin 37 36 bop -24 171 a Fm(Prop)s(osition)38 b(8)49 b Fs(One)34 b(has)1597 446 y Fv(D)1688 381 y Ft(^)1681 398 y FB(\030)1678 473 y(N)1800 446 y Fx(=)1942 378 y(8)17 b Fv(N)2096 342 y Ft(2)p 1942 423 194 4 v 1943 560 a Fw(j)p 1971 480 70 4 v(V)2046 481 y Ft(^)2039 498 y FB(\030)2039 585 y(N)2107 560 y Fw(j)2173 446 y Fv(G)2257 381 y Ft(^)2250 398 y FB(\030)2250 473 y(N)2344 446 y Fv(:)1448 b Fx(\(86\))-24 840 y Fy(7)161 b(Estimates)53 b(for)h(SSMPP)g(obtained)e(b)l(y)h (randomization)-24 1060 y Fx(The)45 b(purp)s(ose)g(of)e(this)i(section) g(is)f(to)g(pro)m(v)m(e)h(Theorems)h(2)d(and)h(3.)77 b(The)45 b(random)f(en)m(vironmen)m(t)i(with)-24 1180 y(distribution)24 b Fw(P)31 b Fx(is)23 b(the)g Fv(\027)6 b Fx(-randomization)23 b(of)f(a)g(SSPP)i(with)f(distribution)2759 1155 y(^)2736 1180 y Fw(P)31 b Fx(and)23 b(densit)m(y)h Fv(\032)p Fx(.)41 b(F)-8 b(urthermore)-24 1300 y(is)103 1275 y(^)81 1300 y Fw(P)158 1264 y FB(c)231 1300 y Fx(the)40 b Fv(\016)449 1315 y FB(c)484 1300 y Fx(-thinning)e(of)1052 1275 y(^)1030 1300 y Fw(P)47 b Fx(with)39 b Fv(\016)1417 1315 y FB(c)1490 1300 y Fx(=)f Fv(\027)6 b Fx(\([)p Fw(\000)p Fv(E)1872 1315 y FB(c)1908 1300 y Fv(;)17 b(E)2024 1315 y FB(c)2059 1300 y Fx(]\).)62 b(Th)m(us)40 b Fv(\032)2516 1315 y FB(c)2589 1300 y Fx(=)e Fv(\016)2746 1315 y FB(c)2781 1300 y Fv(\032)p Fx(.)62 b(W)-8 b(e)39 b(also)g(assume)h(that)f(the)-24 1421 y Fs(Main)c(Hyp)-5 b(othesis)35 b(holds,)f(that)h Fv(\032)1247 1436 y Ft(12)1350 1421 y Fv(<)27 b Fw(1)35 b Fs(and)f(that)h(the)g(dimension)f Fv(d)g Fs(is)h(lar)-5 b(ger)34 b(than)h Fx(1.)-24 1707 y Fl(7.1)136 b(P)l(oin)l(t)46 b(densit)l(y)f(estimates)-24 1892 y Fx(Here)f(w)m(e)g(sho)m(w)g(ho)m(w) f(the)g(ergo)s(dic)g(prop)s(erties)h(of)e(Lemma)i(6)e(com)m(bined)j (with)e(the)g(Main)g(Hyp)s(othesis)-24 2012 y(imply)34 b(\(66\).)-24 2225 y Fm(Prop)s(osition)k(9)49 b Fs(Supp)-5 b(ose)31 b(that)i Fv(\032)1302 2240 y Ft(8)1370 2225 y Fv(<)27 b Fw(1)33 b Fs(and)e(that)i(the)g(Main)f(Hyp)-5 b(othesis)33 b(holds.)43 b(Then)32 b(for)g Fx(1)27 b Fw(\024)i Fv(p)e Fw(\024)h Fx(8)1024 2496 y(lim)1008 2558 y FB(N)7 b Fq("1)1302 2428 y Fv(\032)1352 2443 y FB(c)1404 2428 y Fv(`)p Fx(\()p Fv(C)1553 2443 y FB(N)1620 2428 y Fx(\))p 1231 2473 499 4 v 1241 2554 a(^)1231 2580 y Fv(\030)t Fx(\()p Fv(C)1386 2595 y FB(N)1453 2580 y Fx(\))22 b(+)g Fv(a)1662 2595 y FB(N)1795 2496 y Fx(=)55 b(1)28 b Fv(;)250 b Fs(in)90 b Fv(L)2521 2454 y FB(p)2561 2496 y Fx(\()2648 2470 y(^)2616 2496 y Fw(N)14 b Fv(;)2779 2470 y Fx(^)2756 2496 y Fw(P)2833 2454 y FB(c)2868 2496 y Fx(\))28 b Fv(;)858 b Fx(\(87\))-24 2742 y Fs(wher)-5 b(e)35 b Fv(a)303 2757 y FB(N)398 2742 y Fx(=)27 b(\()p Fv(N)33 b Fw(\000)22 b Fx(1\))836 2706 y FB(d)p Fq(\000)p Ft(1)967 2742 y Fs(.)123 2925 y Fx(W)-8 b(e)33 b(will)g(\014rst)g(pro)m (v)m(e)h(the)f(follo)m(wing)g(criterion.)-24 3138 y Fm(Lemma)39 b(8)49 b Fs(Pr)-5 b(op)g(erty)36 b(\(87\))e(holds)g(if)h(one)f(has,)g (for)h(some)f Fx(0)27 b Fv(<)h(\032)2457 3102 y Fq(0)2508 3138 y Fv(<)g(\032)p Fs(,)1245 3379 y Fx(lim)1228 3442 y FB(N)7 b Fq("1)1441 3379 y Fv(N)1529 3338 y FB(p)1648 3354 y Fx(^)1625 3379 y Fw(P)1719 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Fw(")28 b(1)23 b Fx(b)s(ecause)j(of)d(\(89\).)40 b(F)-8 b(or)23 b(the)i(second,)-24 1165 y(let)44 b(us)g(\014rst)f(note) g(that)g(Lemma)h(6)f(implies)i(that)e(lim)2074 1180 y FB(N)7 b Fq(")p Ft(0)2228 1165 y Fv(f)2276 1180 y FB(N)2344 1165 y Fv(\037)2405 1180 y FB(A)2458 1158 y Fr(c)2458 1204 y(N)2565 1165 y Fx(=)46 b(0)d(holds)3067 1140 y(^)3044 1165 y Fw(P)3121 1129 y FB(c)3157 1165 y Fx(-a.s..)75 b(F)-8 b(urthermore,)-24 1286 y Fw(j)p Fv(f)52 1301 y FB(N)119 1286 y Fv(\037)180 1301 y FB(A)233 1278 y Fr(c)233 1324 y(N)295 1286 y Fw(j)37 b(\024)g Fv(c)516 1250 y Fq(00)596 1286 y Fv(<)g Fw(1)h Fx(uniformly)g(in)h Fv(N)48 b Fx(so)39 b(that)f(the)g(dominated)h(con)m(v)m(ergence)i(theorem)e (assures)g(that)-24 1406 y(lim)112 1421 y FB(N)7 b Fq(")p Ft(0)266 1406 y Fm(E)356 1418 y Ft(^)340 1435 y Fq(P)395 1416 y Fr(c)431 1406 y Fx(\()p Fv(f)517 1421 y FB(N)584 1406 y Fv(\037)645 1421 y FB(A)698 1398 y Fr(c)698 1444 y(N)760 1406 y Fx(\))28 b(=)f(0.)2913 b Ff(2)-24 1574 y Fm(Pro)s(of)46 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3101 y Fv(\030)t Fx(\))2820 3017 y Fo(\014)2820 3076 y(\014)2881 3101 y Fw(\025)f Fx(\()p Fv(\032)3074 3060 y Fq(00)3139 3101 y Fw(\000)23 b Fv(\032)3289 3060 y Fq(0)3312 3101 y Fx(\))p Fv(N)3438 3060 y FB(d)3479 2931 y Fo(\))3604 3101 y Fv(:)-24 3371 y Fx(Hence)34 b(it)f(is)g(su\016cien)m(t)i(to)d(sho)m(w)i(that,)e(for)g Fv(a)c(>)f Fx(0,)1239 3674 y(lim)1222 3737 y FB(N)7 b Fq("1)1408 3674 y Fv(N)1496 3633 y FB(p)1587 3649 y Fx(^)1564 3674 y Fw(P)1658 3503 y Fo( )1737 3589 y(\014)1737 3649 y(\014)1778 3579 y(X)1770 3791 y FB(i)p Fq(2)p FB(I)1872 3802 y Fr(N)1946 3674 y Fv(Y)2003 3689 y FB(i)2031 3589 y Fo(\014)2031 3649 y(\014)2092 3674 y Fw(\025)28 b Fv(aN)2336 3633 y FB(d)2377 3503 y Fo(!)2511 3674 y Fx(=)55 b(0)28 b Fv(:)1073 b Fx(\(91\))-24 3943 y(By)33 b(Cheb)m(yshev)j(inequalit)m (y)-8 b(,)35 b(one)e(has)g(for)f(an)m(y)h(ev)m(en)h Fv(q)e Fw(2)c Fu(N)p Fx(:)767 4221 y(^)744 4246 y Fw(P)838 4076 y Fo( )917 4162 y(\014)917 4221 y(\014)958 4152 y(X)950 4363 y FB(i)p Fq(2)p FB(I)1052 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b(2)p Fv(;)17 b(:)g(:)g(:)e(;)i(q)t Fx(.)62 b(If)38 b Fv(r)3037 4915 y Ft(1)3077 4900 y Fx(\()p Fv(i)p 3115 4916 V Fx(\))g(=)g(0,)i(then)f(use)h(the)-24 5021 y(b)s(ound)1493 5261 y Fm(E)1583 5273 y Ft(^)1567 5290 y Fq(P)1626 5261 y Fx(\()p Fv(Y)1721 5276 y FB(i)1745 5285 y Fp(1)1800 5261 y Fw(\001)17 b(\001)g(\001)d Fv(Y)1989 5276 y FB(i)2013 5284 y Fr(q)2052 5261 y Fx(\))55 b Fw(\024)h Fv(M)2382 5220 y FB(q)2448 5261 y Fv(:)1935 5564 y Fx(38)p eop end %%Page: 39 39 TeXDict begin 39 38 bop -24 171 a Fx(If)28 b Fv(r)113 186 y Ft(1)152 171 y Fx(\()p Fv(i)p 190 187 34 4 v 1 w Fx(\))f Fv(>)h Fx(0,)g(let)g Fv(A)p Fx(\()p Fv(i)p 744 187 V 1 w Fx(\))f(b)s(e)h(the)g(complemen)m(t)i(of)d(the)h(cub)s(e) h(cen)m(tered)g(at)e Fv(i)2698 186 y Ft(1)2766 171 y Fx(with)h(side)h(of)e(length)h(2)p Fv(r)3666 186 y Ft(1)3705 171 y Fx(\()p Fv(i)p 3743 187 V 1 w Fx(\))12 b Fw(\000)g Fx(1.)-24 291 y(Then)34 b Fv(Y)288 306 y FB(i)312 316 y Fr(j)382 291 y Fx(is)f Fw(F)552 306 y FB(A)p Ft(\()p FB(i)p 632 318 25 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Fr(j)3066 4244 y FB(c)3160 4159 y Fo(Q)3254 4262 y FB(j)t Fq(2)p FB(J)3373 4271 y Fp(1)3410 4233 y Fx(\(1)22 b Fw(\000)3623 4207 y Fx(~)3619 4233 y Fv(\016)3666 4179 y FB(s)3699 4189 y Fr(j)3662 4244 y FB(c)3735 4233 y Fx(\))p 466 4315 3308 4 v 1018 4352 a Fo(P)1123 4456 y FB(k)p 1123 4468 39 3 v 1 w Fq(2)p Fj(N)1256 4434 y Fr(J)1290 4449 y Fp(0)1350 4352 y Fo(P)1455 4464 y FB(s)p 1455 4476 33 3 v Fq(2)p Fj(N)1583 4427 y Fr(J)1617 4442 y Fp(1)1583 4485 y(+)1699 4402 y Fx(^)1676 4427 y Fw(P)1753 4346 y Fo(\000)1799 4427 y Fv(W)14 b Fx(\()p Fv(k)p 1943 4443 55 4 v 3 w(;)j(s)p 2041 4443 46 4 v -1 w Fx(\))2124 4346 y Fo(\001)2203 4352 y(Q)2297 4456 y FB(j)t Fq(2)p FB(J)2416 4465 y Fp(0)2475 4401 y Fx(~)2471 4427 y Fv(\016)2518 4373 y FB(k)2555 4383 y Fr(j)2514 4437 y FB(c)2608 4352 y Fo(Q)2702 4456 y FB(j)t Fq(2)p FB(J)2821 4465 y Fp(1)2858 4427 y Fx(\(1)22 b Fw(\000)3071 4401 y Fx(~)3067 4427 y Fv(\016)3114 4373 y FB(s)3147 4383 y Fr(j)3110 4437 y FB(c)3183 4427 y Fx(\))3811 4338 y Fv(:)-24 4669 y Fx(Within)34 b(this,)f(w)m(e)h(can,)f (moreo)m(v)m(er,)h(replace)393 4864 y(^)370 4889 y Fw(P)448 4809 y Fo(\000)504 4863 y Fx(^)494 4889 y Fv(\030)t Fx(\()p Fv(C)649 4904 y FB(r)709 4889 y Fx(+)22 b(2)p Fv(r)s(i)p Fx(\))27 b(=)h Fv(n)17 b(;)33 b(W)14 b Fx(\()p Fv(k)p 1384 4905 55 4 v 2 w(;)j(s)p 1481 4905 46 4 v Fx(\))1565 4809 y Fo(\001)1666 4889 y Fx(=)1820 4864 y(^)1797 4889 y Fw(P)1875 4809 y Fo(\000)1931 4863 y Fx(^)1920 4889 y Fv(\030)5 b Fx(\()p Fv(C)2076 4904 y FB(r)2136 4889 y Fx(+)22 b(2)p Fv(r)s(i)p Fx(\))27 b(=)h Fv(n)17 b Fw(j)g Fv(W)d Fx(\()p Fv(k)p 2796 4905 55 4 v 1 w(;)j(s)p 2892 4905 46 4 v Fx(\))2976 4809 y Fo(\001)3072 4864 y Fx(^)3049 4889 y Fw(P)3127 4809 y Fo(\000)3172 4889 y Fv(W)d Fx(\()p Fv(k)p 3316 4905 55 4 v 3 w(;)j(s)p 3414 4905 46 4 v Fx(\))3498 4809 y Fo(\001)3571 4889 y Fv(:)-24 5109 y Fx(Finally)-8 b(,)33 b(note)g(that)f Fv(W)14 b Fx(\()p Fv(k)p 899 5125 55 4 v 3 w(;)j(s)p 997 5125 46 4 v Fx(\))27 b Fw(2)h(F)1274 5124 y FB(A)1364 5109 y Fx(where)33 b Fv(A)28 b Fx(=)g Fu(R)1922 5073 y FB(d)1984 5109 y Fw(n)2056 5029 y Fo(\000)2102 5109 y Fv(C)2172 5124 y Ft(2)p FB(r)2267 5109 y Fx(+)22 b(2)p Fv(r)s(i)2494 5029 y Fo(\001)2539 5109 y Fx(.)123 5277 y(\(i\))32 b(If)h Fv(E)428 5292 y FB(c)490 5277 y Fx(=)28 b(1,)k(then)h Fv(\016)967 5292 y FB(c)1030 5277 y Fx(=)28 b(1)k(and)1935 5564 y(41)p eop end %%Page: 42 42 TeXDict begin 42 41 bop 313 296 a Fo(X)316 508 y FB(n)p Fq(2)p Fj(N)496 365 y Fx(^)473 391 y Fw(P)551 310 y Fo(\000)607 364 y Fx(^)597 391 y Fv(\030)t Fx(\()p Fv(C)752 406 y FB(r)812 391 y Fx(+)22 b(2)p Fv(r)s(i)p Fx(\))27 b(=)h Fv(n)17 b Fw(j)g Fv(W)d Fx(\()p Fv(k)p 1472 407 55 4 v 2 w(;)j(s)p 1569 407 46 4 v -1 w Fx(\))1652 310 y Fo(\001)1702 364 y Fx(~)1698 391 y Fv(\016)1745 349 y FB(n)1741 415 y(c)1847 391 y Fx(=)2001 365 y(^)1978 391 y Fw(P)2056 310 y Fo(\000)2112 364 y Fx(^)2101 391 y Fv(\030)5 b Fx(\()p Fv(C)2257 406 y FB(r)2317 391 y Fx(+)22 b(2)p Fv(r)s(i)p Fx(\))27 b(=)h(0)17 b Fw(j)g Fv(W)d Fx(\()p Fv(k)p 2968 407 55 4 v 1 w(;)j(s)p 3064 407 46 4 v -1 w Fx(\))3147 310 y Fo(\001)3248 391 y Fw(\024)56 b Fx(1)22 b Fw(\000)h Fv(p)k(;)-24 724 y Fx(where)34 b(the)f(inequalit)m(y)h (follo)m(ws)f(from)f(\(100\))f(and)i(the)f(stationarit)m(y)h(of)2699 699 y(^)2676 724 y Fw(P)9 b Fx(.)43 b(Due)32 b(to)g(the)h(ab)s(o)m(v)m (e)g(computa-)-24 845 y(tions,)g(this)h(inequalit)m(y)g(implies)g (\(103\).)123 1012 y(\(ii\))e(As)434 986 y(~)429 1012 y Fv(\016)472 1027 y FB(c)535 1012 y Fw(\024)c Fv(e)685 976 y Fq(\000)p FB(\016)771 984 y Fr(c)808 1012 y Fx(,)k(w)m(e)i (obtain)e(the)h(follo)m(wing)g(b)s(ound)224 1155 y Fo(X)227 1367 y FB(n)p Fq(2)p Fj(N)407 1224 y Fx(^)385 1250 y Fw(P)462 1169 y Fo(\000)518 1223 y Fx(^)508 1250 y Fv(\030)t Fx(\()p Fv(C)663 1265 y FB(r)723 1250 y Fx(+)22 b(2)p Fv(r)s(i)p Fx(\))28 b(=)f Fv(n)17 b Fw(j)g Fv(W)d Fx(\()p Fv(k)p 1383 1266 55 4 v 2 w(;)j(s)p 1480 1266 46 4 v -1 w Fx(\))1563 1169 y Fo(\001)1614 1223 y Fx(~)1609 1250 y Fv(\016)1656 1209 y FB(n)1652 1274 y(c)1758 1250 y Fw(\024)1914 1224 y Fx(^)1891 1250 y Fw(P)1969 1169 y Fo(\000)2025 1223 y Fx(^)2014 1250 y Fv(\030)5 b Fx(\()p Fv(C)2170 1265 y FB(r)2230 1250 y Fx(+)22 b(2)p Fv(r)s(i)p Fx(\))27 b Fv(<)g(\032)2675 1209 y Fq(0)2699 1250 y Fv(r)2746 1209 y FB(d)2803 1250 y Fw(j)17 b Fv(W)d Fx(\()p Fv(k)p 2992 1266 55 4 v 2 w(;)j(s)p 3089 1266 46 4 v -1 w Fx(\))3172 1169 y Fo(\001)3268 1250 y Fx(+)50 b Fv(e)3439 1209 y Fq(\000)p FB(\016)3525 1217 y Fr(c)3557 1209 y FB(\032)3593 1185 y Fk(0)3615 1209 y FB(r)3649 1185 y Fr(d)3717 1250 y Fv(:)-24 1583 y Fx(Due)33 b(the)g(stationarit)m(y)g(of)1009 1558 y(^)986 1583 y Fw(P)9 b Fx(,)32 b(\(101\))g(and)h(\(102\))e(imply) j(\(103\).)1572 b Ff(2)-24 1751 y Fm(Pro)s(of)37 b(of)g(Theorem)h(2.)44 b Fx(W)-8 b(e)32 b(\014x)h Fv(p)28 b(>)f(p)1552 1766 y FB(c)1587 1751 y Fx(\(2\))32 b(and)g(v)m(erify)i(that)e(\(100\))f (holds)i(if)f Fv(r)j Fx(is)e(large)f(enough.)44 b(This)-24 1871 y(is)e(trivial)f(for)f(a)h(pro)s(cess)h(with)f(a)g(uniform)g(lo)m (w)m(er)h(b)s(ound)f(\(22\))f(on)g(the)i(p)s(oin)m(t)f(densit)m(y)-8 b(.)70 b(F)-8 b(or)40 b(a)g(mixing)-24 1992 y(p)s(oin)m(t)33 b(pro)s(cess)h(satisfying)g(\(23\),)e(one)g(has)382 2207 y(^)360 2232 y Fw(P)437 2122 y Fo(\020)507 2206 y Fx(^)497 2232 y Fv(\030)t Fx(\()p Fv(C)652 2247 y FB(r)690 2232 y Fx(\))27 b Fv(<)h(\032)909 2191 y Fq(0)933 2232 y Fv(r)980 2191 y FB(d)1036 2148 y Fo(\014)1036 2208 y(\014)1069 2232 y Fw(F)1141 2247 y Ft(2)p FB(r)1214 2122 y Fo(\021)1329 2232 y Fw(\024)1485 2207 y Fx(^)1462 2232 y Fw(P)1539 2122 y Fo(\020)1609 2206 y Fx(^)1599 2232 y Fv(\030)t Fx(\()p Fv(C)1754 2247 y FB(r)1792 2232 y Fx(\))g Fv(<)f(\032)2011 2191 y Fq(0)2035 2232 y Fv(r)2082 2191 y FB(d)2122 2122 y Fo(\021)2231 2232 y Fx(+)50 b Fv(r)2404 2191 y FB(d)2461 2232 y Fx(\(2)p Fv(r)s Fx(\))2633 2191 y FB(d)p Fq(\000)p Ft(1)2779 2232 y Fv(h)p Fx(\()p Fv(r)s Fx(\))28 b Fv(;)3247 2207 y Fx(^)3224 2232 y Fw(P)j(\000)23 b Fx(a.s.)17 b Fv(:)-24 2432 y Fx(Due)35 b(to)f(the)h(h)m(yp)s(othesis)i(on)d Fv(h)p Fx(,)i(the)f(second)g(term)g(con)m(v)m(erges)i(to)d(0)h(in)f (the)h(limit)h Fv(r)d Fw(")e(1)p Fx(.)49 b(If)35 b Fv(\032)3583 2396 y Fq(0)3637 2432 y Fv(<)c(\032)p Fx(,)36 b(the)-24 2553 y(\014rst)d(one)g(can)g(b)s(e)g(b)s(ounded)g(b)m(y)h(the)f(Cheb)m (yc)m(hev)j(inequalit)m(y:)263 2930 y(^)240 2956 y Fw(P)9 b Fx(\()366 2929 y(^)356 2956 y Fv(\030)t Fx(\()p Fv(C)511 2971 y FB(r)549 2956 y Fx(\))27 b Fw(\024)h Fv(\032)769 2914 y Fq(0)810 2956 y Fv(r)857 2914 y FB(d)897 2956 y Fx(\))55 b Fw(\024)h Fm(E)1213 2968 y Ft(^)1197 2984 y Fq(P)1272 2785 y Fo( )1351 2781 y(\014)1351 2841 y(\014)1351 2901 y(\014)1351 2961 y(\014)1351 3020 y(\014)1404 2862 y Fx(^)1394 2888 y Fv(\030)5 b Fx(\()p Fv(C)1550 2903 y FB(r)1587 2888 y Fx(\))p 1394 2933 232 4 v 1397 3024 a Fv(`)p Fx(\()p Fv(C)1546 3039 y FB(r)1584 3024 y Fx(\))1657 2956 y Fw(\000)23 b Fv(\032)1807 2781 y Fo(\014)1807 2841 y(\014)1807 2901 y(\014)1807 2961 y(\014)1807 3020 y(\014)1868 2956 y Fv(>)28 b(\032)22 b Fw(\000)h Fv(\032)2194 2914 y Fq(0)2217 2785 y Fo(!)2352 2956 y Fw(\024)2593 2888 y Fx(1)p 2494 2933 246 4 v 2494 3024 a Fv(\032)g Fw(\000)f Fv(\032)2716 2995 y Fq(0)2767 2820 y Fo(Z)2906 2930 y Fx(^)2883 2956 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))3168 2781 y Fo(\014)3168 2841 y(\014)3168 2901 y(\014)3168 2961 y(\014)3168 3020 y(\014)3221 2862 y Fx(^)3211 2888 y Fv(\030)t Fx(\()p Fv(C)3366 2903 y FB(r)3404 2888 y Fx(\))p 3211 2933 232 4 v 3214 3024 a Fv(`)p Fx(\()p Fv(C)3363 3039 y FB(r)3401 3024 y Fx(\))3474 2956 y Fw(\000)22 b Fv(\032)3623 2781 y Fo(\014)3623 2841 y(\014)3623 2901 y(\014)3623 2961 y(\014)3623 3020 y(\014)3701 2956 y Fv(:)-24 3257 y Fx(By)h(Lemma)g(6,)i(the)e(expression)h(on)f(the)g (r.h.s.)41 b(can)22 b(b)s(e)h(made)g(arbitrarily)g(small)g(b)m(y)g(c)m (ho)s(osing)g Fv(r)i Fx(su\016cien)m(tly)-24 3378 y(large,)31 b(th)m(us)f(implying)h(that)f(\(100\))e(is)j(satis\014ed)g(for)e Fv(r)j Fx(su\016cien)m(tly)h(large.)42 b(In)30 b(conclusion,)i(due)e (to)f(Lemma)-24 3498 y(9,)k(\(98\))f(holds)h(for)f Fv(r)j Fx(large)e(enough.)44 b(W)-8 b(e)33 b(\014x)g(suc)m(h)h(a)e(v)-5 b(alue)33 b Fv(r)i Fx(satisfying)f(\(98\))e(and)h(call)f(it)h Fv(r)3476 3513 y FB(p)3516 3498 y Fx(.)123 3677 y(Consider)27 b(the)g(v)-5 b(ariables)27 b(\()p Fv(\033)1176 3692 y FB(j)1213 3677 y Fx(\))1251 3697 y FB(j)t Fq(2)p Fj(Z)1375 3678 y Fr(d)1441 3677 y Fx(de\014ned)h(for)e Fv(r)1958 3692 y Ft(1)2025 3677 y Fx(=)h Fv(r)2172 3692 y FB(p)2212 3677 y Fv(;)17 b(r)2300 3692 y Ft(2)2367 3677 y Fx(=)27 b(2)p Fv(r)2563 3692 y FB(p)2629 3677 y Fx(and)f(c)m(ho)s(ose)i Fv(r)3160 3692 y FB(c)3222 3677 y Fx(=)g(\()p Fv(d)9 b Fx(+)g(8\))3606 3613 y Fp(1)p 3605 3625 31 3 v 3605 3667 a(2)3650 3677 y Fv(r)3694 3692 y FB(p)3734 3677 y Fx(.)41 b(This)-24 3813 y(assures)d(that,)f(if)f(neigh)m(b)s(oring)h (sites)h Fv(j)k Fx(and)36 b Fv(j)1726 3777 y Fq(0)1785 3813 y Fx(in)h Fu(Z)1969 3777 y FB(d)2046 3813 y Fx(ha)m(v)m(e)g Fv(\033)2329 3828 y FB(j)2366 3813 y Fx(\()2414 3787 y(^)2404 3813 y Fv(\030)5 b Fx(\))34 b(=)f Fv(\033)2688 3828 y FB(j)2721 3809 y Fk(0)2748 3813 y Fx(\()2796 3787 y(^)2786 3813 y Fv(\030)t Fx(\))h(=)g(1,)j(then)f Fv(C)3423 3828 y FB(r)3455 3836 y Fr(p)3520 3813 y Fx(+)24 b(2)p Fv(j)6 b(r)3759 3828 y FB(p)3835 3813 y Fx(and)-24 3934 y Fv(C)46 3949 y FB(r)78 3957 y Fr(p)135 3934 y Fx(+)16 b(2)p Fv(j)322 3897 y Fq(0)345 3934 y Fv(r)389 3949 y FB(p)458 3934 y Fx(con)m(tain)30 b(eac)m(h)h(a)f(p)s(oin)m(t)f(and)h (these)h(p)s(oin)m(ts)g(are)e(separated)i(b)m(y)g(a)e(distance)i(less)g (than)f Fv(r)3701 3949 y FB(c)3736 3934 y Fx(.)42 b(Tw)m(o)-24 4070 y(neigh)m(b)s(oring)26 b(sites)g Fv(j)k Fx(and)25 b Fv(j)1008 4034 y Fq(0)1056 4070 y Fx(in)g Fu(Z)1228 4034 y FB(d)1293 4070 y Fx(suc)m(h)h(that)f Fv(\033)1764 4085 y FB(j)1801 4070 y Fx(\()1849 4044 y(^)1839 4070 y Fv(\030)t Fx(\))j(=)f Fv(\033)2110 4085 y FB(j)2143 4066 y Fk(0)2169 4070 y Fx(\()2217 4044 y(^)2207 4070 y Fv(\030)5 b Fx(\))27 b(=)h(1)c(de\014ne)i(a)f(b)s(ond)f(of)g(the)h 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Fx(\()p Fv(dy)t Fx(\))2519 423 y Fo(Z)2573 649 y FB(S)2616 658 y Fk(\000)p Fr(y)2701 649 y FB(A)2785 532 y Fx(^)2775 559 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))p Fv(f)11 b Fx(\()p Fv(S)3161 574 y FB(y)3202 559 y Fv(\030)5 b(;)17 b(S)3354 574 y FB(x)3397 559 y Fv(\030)5 b Fx(\))27 b Fv(:)233 b Fx(\(104\))-24 849 y(The)34 b(an)m(tisymmetry)h(of)d Fv(h)p Fx(\()p Fv(\030)5 b(;)17 b(\030)1143 813 y Fq(0)1165 849 y Fx(\))32 b(and)h(the)g(iden)m (tit)m(y)i(ab)s(o)m(v)m(e)e(imply)162 981 y Fo(Z)278 1117 y Fw(P)347 1132 y Ft(0)387 1117 y Fx(\()p Fv(d\030)5 b Fx(\))579 981 y Fo(Z)633 1207 y Fj(R)681 1188 y Fr(d)748 1091 y Fx(^)738 1117 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))17 b Fv(h)p Fx(\()p Fv(\030)5 b(;)17 b(S)1230 1132 y FB(x)1273 1117 y Fv(\030)5 b Fx(\))27 b(=)1565 1050 y(1)p 1499 1094 V 1499 1185 a Fv(\032N)1637 1157 y FB(d)1705 981 y Fo(Z)1821 1117 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))2090 981 y Fo(Z)2145 1207 y FB(C)2195 1218 y Fr(N)2284 1091 y Fx(^)2274 1117 y Fv(\030)t Fx(\()p Fv(dy)t Fx(\))2517 981 y Fo(Z)2571 1207 y Fj(R)2619 1188 y Fr(d)2655 1207 y Fq(n)p FB(C)2740 1218 y Fr(N)2829 1091 y Fx(^)2819 1117 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))17 b Fv(h)p Fx(\()p Fv(S)3219 1132 y FB(y)3260 1117 y Fv(\030)5 b(;)17 b(S)3412 1132 y FB(x)3455 1117 y Fv(\030)5 b Fx(\))17 b Fv(:)185 b Fx(\(105\))-24 1413 y(Let)33 b(us)g(split)g(the)g(last)g(in)m(tegral) g(in)m(to)g(t)m(w)m(o)g(in)m(tegrals)g(o)m(v)m(er)h Fu(R)2257 1377 y FB(d)2319 1413 y Fw(n)22 b Fv(C)2461 1441 y FB(N)7 b Ft(+)2579 1385 y Fq(p)p 2638 1385 64 3 v 56 x FB(N)2737 1413 y Fx(and)33 b(o)m(v)m(er)g Fv(C)3205 1441 y FB(N)7 b Ft(+)3323 1385 y Fq(p)p 3382 1385 V 56 x FB(N)3471 1413 y Fw(n)22 b Fv(C)3613 1428 y FB(N)3680 1413 y Fx(.)43 b(Using)-24 1533 y(\(104\))32 b(again,)926 1868 y(1)p 860 1913 180 4 v 860 2004 a Fv(\032N)998 1975 y FB(d)1066 1761 y Fo(\014)1066 1821 y(\014)1066 1881 y(\014)1066 1940 y(\014)1066 2000 y(\014)1099 1800 y(Z)1216 1935 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))1485 1800 y Fo(Z)1539 2025 y FB(C)1589 2036 y Fr(N)1678 1909 y Fx(^)1668 1935 y Fv(\030)t Fx(\()p Fv(dy)t Fx(\))1911 1800 y Fo(Z)1965 2025 y Fj(R)2013 2006 y Fr(d)2049 2025 y Fq(n)p FB(C)2134 2050 y Fr(N)g Fp(+)2234 2008 y Fk(p)p 2285 2008 54 3 v 2285 2050 a Fr(N)2373 1909 y Fx(^)2363 1935 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))p Fv(h)p Fx(\()p Fv(S)2746 1950 y FB(y)2788 1935 y Fv(\030)g(;)17 b(S)2940 1950 y FB(x)2983 1935 y Fv(\030)5 b Fx(\))3069 1761 y Fo(\014)3069 1821 y(\014)3069 1881 y(\014)3069 1940 y(\014)3069 2000 y(\014)927 2229 y Fw(\024)1060 2093 y Fo(Z)1176 2229 y Fw(P)1245 2244 y Ft(0)1285 2229 y Fx(\()p Fv(d\030)g Fx(\))1477 2093 y Fo(Z)1531 2319 y Fj(R)1579 2300 y Fr(d)1615 2319 y Fq(n)p FB(C)1700 2301 y Fk(p)p 1750 2301 V 42 x Fr(N)1839 2202 y Fx(^)1829 2229 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))2058 2148 y Fo(\000)2104 2229 y Fw(j)p Fv(k)s Fx(\()p Fv(\030)g(;)17 b(S)2376 2244 y FB(x)2419 2229 y Fv(\030)5 b Fx(\))p Fw(j)21 b Fx(+)h Fw(j)p Fv(k)s Fx(\()p Fv(S)2832 2244 y FB(x)2875 2229 y Fv(\030)5 b(;)17 b(\030)5 b Fx(\))p Fw(j)3081 2148 y Fo(\001)3153 2229 y Fv(;)-24 2525 y Fx(whic)m(h)38 b(con)m(v)m(erges)h(to)d(zero)g(as)h Fv(N)45 b Fw(!)33 b(1)j Fx(b)m(y)h(the)g(dominated)g(con)m(v)m(ergence)i (theorem.)56 b(The)37 b(same)g(holds)-24 2645 y(for)878 2760 y(1)p 813 2805 180 4 v 813 2896 a Fv(\032N)951 2867 y FB(d)1019 2653 y Fo(\014)1019 2713 y(\014)1019 2773 y(\014)1019 2833 y(\014)1019 2892 y(\014)1052 2692 y(Z)1168 2828 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))1437 2692 y Fo(Z)1491 2918 y FB(C)1541 2929 y Fr(N)1630 2801 y Fx(^)1620 2828 y Fv(\030)g Fx(\()p Fv(dy)t Fx(\))1864 2692 y Fo(Z)1917 2918 y FB(C)1967 2942 y Fr(N)g Fp(+)2067 2900 y Fk(p)p 2118 2900 54 3 v 2118 2942 a Fr(N)2175 2918 y Fq(n)p FB(C)2260 2929 y Fr(N)2349 2801 y Fx(^)2339 2828 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))17 b Fv(h)p Fx(\()p Fv(S)2740 2843 y FB(y)2781 2828 y Fv(\030)5 b(;)17 b(S)2933 2843 y FB(x)2976 2828 y Fv(\030)5 b Fx(\))3062 2653 y Fo(\014)3062 2713 y(\014)3062 2773 y(\014)3062 2833 y(\014)3062 2892 y(\014)3139 2828 y Fv(;)-24 3089 y Fx(since,)34 b(due)g(to)e(\(104\),)g(it)g(can)h (b)s(e)g(b)s(ounded)g(b)m(y)570 3289 y(1)p 505 3334 180 4 v 505 3425 a Fv(\032N)643 3396 y FB(d)711 3221 y Fo(Z)827 3357 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))1096 3221 y Fo(Z)1150 3447 y FB(C)1200 3471 y Fr(N)g Fp(+)1300 3429 y Fk(p)p 1351 3429 54 3 v 1351 3471 a Fr(N)1408 3447 y Fq(n)p FB(C)1493 3458 y Fr(N)1583 3330 y Fx(^)1572 3357 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))1819 3221 y Fo(Z)1874 3447 y Fj(R)1922 3428 y Fr(d)1989 3330 y Fx(^)1979 3357 y Fv(\030)t Fx(\()p Fv(dy)t Fx(\))2205 3276 y Fo(\000)2249 3357 y Fw(j)p Fv(k)s Fx(\()p Fv(S)2429 3372 y FB(y)2471 3357 y Fv(\030)g(;)17 b(S)2623 3372 y FB(x)2666 3357 y Fv(\030)5 b Fx(\))p Fw(j)21 b Fx(+)h Fw(j)p Fv(k)s Fx(\()p Fv(S)3079 3372 y FB(x)3122 3357 y Fv(\030)5 b(;)17 b(S)3274 3372 y FB(y)3315 3357 y Fv(\030)5 b Fx(\))p Fw(j)3429 3276 y Fo(\001)582 3671 y Fx(=)695 3603 y(\()p Fv(N)33 b Fx(+)942 3519 y Fw(p)p 1025 3519 89 4 v 84 x Fv(N)10 b Fx(\))1151 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b Fx(\))27 b Fv(;)212 b Fw(8)28 b Fv(\030)k Fw(2)c(N)15 b Fv(;)44 b Fw(8)28 b Fv(x)g Fw(2)2784 4280 y Fx(^)2774 4306 y Fv(\030)k(:)-24 4525 y Fx(The)i(ab)s(o)m(v)m(e)f(remark)h(together)e(with)i(\(6\))e (giv)m(es)634 4793 y Fw(P)703 4808 y Ft(0)742 4793 y Fx(\(\000)841 4808 y Ft(0)881 4793 y Fx(\))55 b(=)1116 4726 y(1)p 1115 4770 51 4 v 1115 4862 a Fv(\032)1192 4658 y Fo(Z)1308 4793 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))1577 4658 y Fo(Z)1632 4883 y FB(C)1682 4892 y Fp(1)1747 4767 y Fx(^)1737 4793 y Fv(\030)g Fx(\()p Fv(dx)p Fx(\))p Fv(\037)2028 4808 y Ft(\000)2072 4817 y Fp(0)2111 4793 y Fx(\()p Fv(S)2209 4808 y FB(x)2252 4793 y Fv(\030)g Fx(\))55 b(=)2535 4726 y(1)p 2534 4770 V 2534 4862 a Fv(\032)2611 4658 y Fo(Z)2667 4883 y Ft(\000)2731 4793 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))3010 4767 y(^)3000 4793 y Fv(\030)t Fx(\()p Fv(C)3155 4808 y Ft(1)3194 4793 y Fx(\))32 b(.)-24 5074 y(Comparing)40 b(with)g(\(4\),)h(this)f(yields) h Fw(P)1456 5089 y Ft(0)1496 5074 y Fx(\(\000)1595 5089 y Ft(0)1634 5074 y Fx(\))e(=)h(1)f(if)g Fw(P)8 b Fx(\(\000\))39 b(=)h(1.)63 b(Recipro)s(cally)-8 b(,)42 b(alw)m(a)m(ys)f(due)f(to)f (\(4\),)i(if)-24 5195 y Fw(P)45 5210 y Ft(0)85 5195 y Fx(\(\000)184 5210 y Ft(0)223 5195 y Fx(\))28 b(=)g(1,)i(one)f(gets)883 5168 y(^)873 5195 y Fv(\030)t Fx(\()p Fv(C)1028 5210 y Ft(1)1067 5195 y Fx(\))f(=)f(0)j(for)f Fw(P)8 b Fx({almost)30 b(all)f Fv(\030)j Fw(2)d(N)h(n)16 b Fx(\000,)30 b(and)g(b)m(y)g (translation)g(in)m(v)-5 b(ariance)31 b Fv(\030)h Fx(=)27 b(0)-24 5315 y(for)32 b Fw(P)8 b Fx({almost)34 b(all)e Fv(\030)g Fw(2)c(N)37 b(n)22 b Fx(\000,)32 b(th)m(us)i(implying)g(that) e Fw(P)8 b Fx(\(\000\))28 b(=)g(1.)1935 5564 y(44)p eop end %%Page: 45 45 TeXDict begin 45 44 bop -24 171 a Fx(\(iii\))35 b(Let)g(us)h(supp)s (ose)g(that)f Fw(P)1125 186 y Ft(0)1164 171 y Fx(\()p Fv(A)p Fx(\))d(=)f Fw(P)1521 186 y Ft(0)1561 171 y Fx(\()p Fv(B)5 b Fx(\))32 b Fv(>)f Fx(0.)50 b(and)35 b(set)h(\000)31 b(:=)2555 96 y Fo(S)2638 200 y FB(x)p Fq(2)p Fj(R)2773 181 y Fr(d)2829 171 y Fv(S)2889 186 y FB(x)2933 171 y Fv(B)5 b Fx(.)50 b(This)36 b(is)g(a)e(translation-)-24 291 y(in)m(v)-5 b(arian)m(t)30 b(Borel)g(subset)h(of)d Fw(N)44 b Fx(\(see)30 b(Lemma)g(10\))f(and)h Fv(B)i Fw(\032)c Fx(\000)15 b Fw(\\)g(N)2506 306 y Ft(0)2575 291 y Fw(\032)28 b Fv(A)p Fx(.)42 b(In)30 b(particular,)g Fw(P)8 b Fx(\(\000\))28 b Fw(2)g(f)p Fx(0)p Fv(;)17 b Fx(1)p Fw(g)-24 411 y Fx(b)m(y)34 b(the)f(ergo)s(dicit)m(y)g(of)f Fw(P)8 b Fx(.)44 b(Since)1073 631 y Fv(\037)1134 646 y FB(B)1195 631 y Fx(\()p Fv(S)1293 646 y FB(y)1335 631 y Fv(\030)5 b Fx(\))27 b Fw(\024)h Fv(\037)1614 646 y Ft(\000)1662 631 y Fx(\()p Fv(\030)5 b Fx(\))27 b Fv(;)212 b Fw(8)28 b Fv(\030)k Fw(2)c(N)15 b Fv(;)44 b Fw(8)28 b Fv(y)j Fw(2)d Fu(R)2800 590 y FB(d)2868 631 y Fv(;)-24 851 y Fx(if)33 b(follo)m(ws)g(from)f(\(6\))h(that)685 1120 y Fw(P)754 1135 y Ft(0)794 1120 y Fx(\()p Fv(B)5 b Fx(\))55 b(=)1146 1052 y(1)p 1146 1097 51 4 v 1146 1188 a Fv(\032)1222 984 y Fo(Z)1278 1210 y Fq(N)1366 1120 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))1635 984 y Fo(Z)1690 1210 y FB(C)1740 1219 y Fp(1)1805 1094 y Fx(^)1795 1120 y Fv(\030)g Fx(\()p Fv(dy)t Fx(\))17 b Fv(\037)2100 1135 y FB(B)2159 1120 y Fx(\()p Fv(S)2257 1135 y FB(y)2298 1120 y Fv(\030)5 b Fx(\))55 b Fw(\024)2582 1052 y Fx(1)p 2582 1097 V 2582 1188 a Fv(\032)2658 984 y Fo(Z)2714 1210 y Ft(\000)2779 1120 y Fw(P)8 b Fx(\()p Fv(\030)d Fx(\))2990 1094 y(^)2980 1120 y Fv(\030)20 b Fx(\()p Fv(C)3151 1135 y Ft(1)3191 1120 y Fx(\))27 b Fv(:)-24 1402 y Fx(Therefore,)32 b Fw(P)8 b Fx(\(\000\))28 b(=)g(0)i(w)m(ould)h (imply)g(that)f Fw(P)1699 1417 y Ft(0)1739 1402 y Fx(\()p Fv(B)5 b Fx(\))27 b(=)h(0,)i(in)h(con)m(tradiction)g(with)f(our)g (assumption.)45 b(Th)m(us)-24 1522 y Fw(P)8 b Fx(\(\000\))28 b(=)g(1.)43 b(But)33 b(\000)22 b Fw(\\)h(N)889 1537 y Ft(0)955 1522 y Fw(\032)29 b Fv(A)p Fx(,)j(therefore)h(the)g(statemen)m (t)h(follo)m(ws)g(from)e(\(ii\).)-24 1666 y(\(iv\))h(The)h(thesis)g (follo)m(ws)f(b)m(y)h(observing)g(that)e(\(6\))g(implies)196 1979 y Fm(E)270 1994 y Fq(P)319 2003 y Fp(0)358 1899 y Fo(\000)464 1855 y FB(k)420 1885 y Fo(Y)422 2095 y FB(j)t Ft(=1)574 1953 y Fx(^)564 1979 y Fv(\030)t Fx(\()p Fv(A)722 1994 y FB(j)759 1979 y Fx(\))814 1899 y Fo(\001)903 1979 y Fx(=)1034 1912 y(1)p 1034 1957 V 1034 2048 a Fv(\032)1127 1844 y Fo(Z)1182 2069 y Fq(N)1271 1979 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))1540 1844 y Fo(Z)1594 2069 y FB(C)1644 2078 y Fp(1)1710 1953 y Fx(^)1700 1979 y Fv(\030)t Fx(\()p Fv(dx)p Fx(\))1990 1855 y FB(k)1946 1885 y Fo(Y)1948 2095 y FB(j)t Ft(=1)2100 1953 y Fx(^)2090 1979 y Fv(\030)t Fx(\()p Fv(A)2248 1994 y FB(j)2307 1979 y Fx(+)22 b Fv(x)p Fx(\))45 b Fw(\024)2675 1912 y Fx(1)p 2674 1957 V 2674 2048 a Fv(\032)2751 1844 y Fo(Z)2807 2069 y Fq(N)2895 1979 y Fw(P)8 b Fx(\()p Fv(d\030)d Fx(\))3157 1953 y(^)3147 1979 y Fv(\030)t Fx(\()p Fv(C)3302 1994 y Ft(1)3341 1979 y Fx(\))3440 1855 y FB(k)3396 1885 y Fo(Y)3398 2095 y FB(j)t Ft(=1)3550 1953 y Fx(^)3539 1979 y Fv(\030)g Fx(\()3651 1954 y(~)3625 1979 y Fv(A)3698 1994 y FB(j)3734 1979 y Fx(\))-24 2313 y(and)33 b(b)m(y)h(applying)f(the)g(estimate)h Fv(a)1310 2328 y Ft(1)1366 2313 y Fw(\001)17 b(\001)g(\001)e Fv(a)1550 2328 y FB(k)r Ft(+1)1710 2313 y Fw(\024)29 b Fv(c)p Fx(\()p Fv(k)c Fx(+)d(1\))17 b(\()p Fv(a)2263 2272 y FB(k)r Ft(+1)2263 2337 y(1)2417 2313 y Fx(+)22 b Fw(\001)17 b(\001)g(\001)k Fx(+)h Fv(a)2803 2272 y FB(k)r Ft(+1)2803 2341 y FB(k)r Ft(+1)2936 2313 y Fx(\),)32 b Fv(a)3084 2328 y Ft(1)3124 2313 y Fv(;)17 b(:)g(:)g(:)f(;)h(a)3394 2328 y FB(k)r Ft(+1)3554 2313 y Fw(\025)28 b Fx(0.)183 b Ff(2)-24 2557 y Fm(Lemma)39 b(10)49 b Fs(L)-5 b(et)36 b Fv(A)27 b Fw(2)h(B)s Fx(\()p Fw(N)1085 2572 y Ft(0)1125 2557 y Fx(\))p Fs(.)45 b(Then)1492 2482 y Fo(S)1575 2586 y FB(x)p Fq(2)p Fj(R)1710 2567 y Fr(d)1766 2557 y Fv(S)1826 2572 y FB(x)1870 2557 y Fv(A)28 b Fw(2)g(B)s Fx(\()p Fw(N)15 b Fx(\))p Fs(.)-24 2743 y Fm(Pro)s(of.)49 b Fx(Let)30 b(us)g(in)m(tro)s(duce)h(the)f(follo)m(wing)g(lexicographic)h(ordering) f(on)g Fu(R)2799 2707 y FB(d)2839 2743 y Fx(:)42 b Fv(x)28 b Fw(\036)g Fv(y)33 b Fx(if)d(and)g(only)g(if)f(either)-24 2864 y Fw(j)p Fv(x)p Fw(j)k Fv(<)g Fw(j)p Fv(y)t Fw(j)h Fx(or)i Fw(j)p Fv(x)p Fw(j)c Fx(=)h Fw(j)p Fv(y)t Fw(j)i Fx(and)g(there)i(is)f Fv(k)s Fx(,)h(1)32 b Fw(\024)i Fv(k)i Fw(\024)e Fv(d)p Fx(,)i(suc)m(h)h(that)f Fv(x)2550 2828 y Ft(\()p FB(k)r Ft(\))2681 2864 y Fv(<)d(y)2842 2828 y Ft(\()p FB(k)r Ft(\))2974 2864 y Fx(and)j Fv(x)3222 2828 y Ft(\()p FB(l)q Ft(\))3336 2864 y Fx(=)d Fv(y)3497 2828 y Ft(\()p FB(l)q Ft(\))3613 2864 y Fx(for)i Fv(l)g(<)e(k)-24 2984 y Fx(\(here)c Fv(x)276 2948 y Ft(\()p FB(k)r Ft(\))403 2984 y Fx(is)f(the)h Fv(k)s Fx(-th)f(comp)s(onen)m(t)h(of)f(the)h(v)m (ector)g Fv(x)p Fx(\).)43 b(Giv)m(en)2371 2958 y(^)2361 2984 y Fv(\030)32 b Fw(2)2562 2959 y Fx(^)2530 2984 y Fw(N)14 b Fx(,)30 b(one)e(can)h(then)g(order)f(the)h(supp)s(ort)-24 3105 y(of)98 3078 y(^)87 3105 y Fv(\030)37 b Fx(according)c(to)g Fw(\036)p Fx(:)680 3406 y(supp)q(\()929 3380 y(^)919 3406 y Fv(\030)t Fx(\))55 b(=)1191 3236 y Fo(\()1271 3338 y Fw(f)p Fv(y)1369 3353 y Ft(1)1408 3338 y Fx(\()1456 3312 y(^)1446 3338 y Fv(\030)t Fx(\))p Fv(;)17 b(y)1623 3353 y Ft(2)1662 3338 y Fx(\()1710 3312 y(^)1700 3338 y Fv(\030)t Fx(\))p Fv(;)g(:)g(:)g(:)f(;)h(y)2052 3353 y FB(N)2118 3338 y Fx(\()2166 3312 y(^)2156 3338 y Fv(\030)5 b Fx(\))p Fw(g)129 b Fx(if)33 b Fv(N)38 b Fx(:=)2768 3312 y(^)2758 3338 y Fv(\030)t Fx(\()p Fu(R)2915 3302 y FB(d)2955 3338 y Fx(\))28 b Fv(<)g Fw(1)f Fv(;)1271 3482 y Fw(f)p Fv(y)1369 3497 y FB(j)1405 3482 y Fx(\()1453 3456 y(^)1443 3482 y Fv(\030)t Fx(\))p Fw(g)1578 3497 y FB(j)t Fq(2)p Fj(N)1706 3506 y Fp(+)2421 3482 y Fx(otherwise)34 b Fv(;)-24 3733 y Fx(where)g Fv(y)306 3748 y FB(j)370 3733 y Fw(\036)28 b Fv(y)523 3748 y FB(k)598 3733 y Fx(whenev)m(er)35 b Fv(j)f(<)27 b(k)s Fx(.)44 b(F)-8 b(or)31 b(an)m(y)j Fv(n)28 b Fw(2)g Fu(N)p Fx(,)k(let)h Fv(x)2194 3748 y FB(n)2269 3733 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Fx(:)31 b Fu(R)2357 4864 y FB(d)2421 4900 y Fw(\002)24 b(N)45 b(!)31 b(N)48 b Fx(giv)m(en)36 b(b)m(y)f Fv(F)14 b Fx(\()p Fv(x;)j(\030)5 b Fx(\))30 b(=)h Fv(S)3801 4915 y FB(x)3845 4900 y Fv(\030)39 b Fx(is)-24 5020 y(ev)m(en)c(con)m(tin)m (uous.)45 b(Hence)34 b(w)m(e)f(conclude)h(that)806 5261 y Fv(H)887 5276 y FB(n)962 5261 y Fx(:)28 b Fw(N)42 b(!)27 b(N)1350 5276 y Ft(0)1417 5261 y Fv(;)212 b(H)1737 5276 y FB(n)1783 5261 y Fx(\()p Fv(\030)5 b Fx(\))55 b(:=)g Fv(F)2197 5180 y Fo(\000)2259 5261 y Fv(x)2314 5276 y FB(n)2362 5261 y Fx(\()2410 5235 y(^)2400 5261 y Fv(\030)t Fx(\))p Fv(;)17 b(\030)2593 5180 y Fo(\001)2694 5261 y Fx(=)55 b Fv(S)2885 5290 y FB(x)2925 5298 y Fr(n)2967 5290 y Ft(\()3001 5273 y(^)2994 5290 y FB(\030)s Ft(\))3060 5261 y Fv(\030)32 b(;)1935 5564 y Fx(45)p eop end %%Page: 46 46 TeXDict begin 46 45 bop -24 171 a Fx(is)38 b(a)e(Borel)h(function.)56 b(Its)37 b(restriction)1503 145 y(^)1478 171 y Fv(H)1559 186 y FB(n)1640 171 y Fx(:)e Fw(N)1784 186 y Ft(0)1858 171 y Fw(!)f(N)2074 186 y Ft(0)2150 171 y Fx(is)j(then)g(also)g(a)f (Borel)h(function.)56 b(No)m(w)38 b(giv)m(en)f(a)-24 291 y(Borel)h(subset)i Fv(A)e Fx(of)f Fw(N)854 306 y Ft(0)893 291 y Fx(,)j(w)m(e)e(conclude)i(that)d(\010\()p Fv(A)p Fx(\))h(:=)2127 216 y Fo(S)2210 243 y Fq(1)2210 320 y FB(n)p Ft(=1)2389 266 y Fx(^)2364 291 y Fv(H)2453 255 y Fq(\000)p Ft(1)2445 316 y FB(n)2547 291 y Fx(\()p Fv(A)p Fx(\))g(is)g(a)g(Borel)g(subset)h(in)f Fw(N)3693 306 y Ft(0)3732 291 y Fx(.)60 b(One)-24 411 y(can)33 b(c)m(hec)m(k)i(that)858 532 y(\010\()p Fv(A)p Fx(\))55 b(=)h Fw(f)p Fv(\030)48 b Fx(:)c Fv(\030)60 b Fx(=)55 b Fv(S)1770 547 y FB(x)1814 532 y Fv(\021)64 b Fx(for)32 b(some)i Fv(\021)d Fw(2)d Fv(A)33 b Fx(and)f Fv(x)d Fw(2)35 b Fx(^)-56 b Fv(\021)20 b Fw(g)p Fv(:)-24 699 y Fx(Since)34 b Fw(N)313 714 y Ft(0)385 699 y Fx(is)f(a)f(Borel)h(subset)h(of)e Fw(N)15 b Fx(,)32 b(it)h(follo)m(ws)g(that)f(\010\()p Fv(A)p Fx(\))h(is)g(a)g(Borel)f(subset)i(of)f Fw(N)46 b Fx(as)33 b(is)g Fv(H)3554 658 y Fq(\000)p Ft(1)3546 723 y(1)3648 618 y Fo(\000)3710 699 y Fx(\010\()p Fv(A)p Fx(\))3946 618 y Fo(\001)-24 819 y Fx(since)h Fv(H)296 834 y Ft(1)368 819 y Fx(is)f(a)g(Borel)f(function.)44 b(The)34 b(iden)m(tit)m(y)1399 1035 y Fv(H)1488 994 y Fq(\000)p Ft(1)1480 1059 y(1)1582 954 y Fo(\000)1644 1035 y Fx(\010\()p Fv(A)p Fx(\))1880 954 y Fo(\001)1981 1035 y Fx(=)2143 940 y Fo([)2113 1160 y FB(x)p Fq(2)p Fj(R)2248 1141 y Fr(d)2300 1035 y Fv(S)2360 1050 y FB(x)2404 1035 y Fv(A)f Fx(,)-24 1334 y(no)m(w)h(completes)g(the)f(pro)s(of.)2870 b Ff(2)-24 1665 y Fy(References)-24 1884 y Fx([AHL])50 b(V.)31 b(Am)m(b)s(egoak)-5 b(ar,)32 b(B.)g(I.)g(Halp)s(erin,)g(J.)g (S.)g(Langer,)g Fn(Hopping)g(Conductivit)m(y)i(in)e(Disordered)g(Sys-) 177 2004 y(tems)p Fx(,)i(Ph)m(ys,)g(Rev)f Fm(B)k(4)p Fx(,)c(2612-2620)d(\(1971\).)-24 2203 y([BRSW])50 b(J.)44 b(Bellissard,)49 b(R.)44 b(Reb)s(olledo,)j(D.)d(Sp)s(ehner,)k(W.)d(v)m (on)g(W)-8 b(aldenfels,)48 b(in)d(preparation;)50 b(same)177 2323 y(authors,)30 b Fn(The)g(quan)m(tum)h(\015o)m(w)f(of)e(electronic) j(transp)s(ort)e(I:)h(the)f(\014nite)h(v)m(olume)h(case)p Fx(,)g(preprin)m(t)f(2002,)177 2444 y Fa(mp)p 285 2444 31 4 v 37 w(arc/02-212)-24 2642 y Fx([BHZ])49 b(J.)39 b(Bellissard,)k(D.)38 b(Hermann,)k(M.)e(Zarrouati,)f Fn(Hull)h(of)f(Ap)s(erio)s(dic)g(Solids)h(and)f(Gap)g(Lab)s(elling)177 2763 y(Theorems)p Fx(,)51 b(in)46 b(Directions)h(in)f(Mathematical)h (Quasicrystals,)52 b(M.B.)46 b(Baak)m(e)h(and)f(R.V.)g(Mo)s(o)s(dy)177 2883 y(Eds.,)34 b(AMS)f(Pro)m(vidence,)i(CRM)e(Monograph)g(Series,)h(V) -8 b(olume)33 b(13,)f(207-259,)f(\(2000\).)-24 3082 y([Bil])50 b(P)-8 b(.)32 b(Billingsley)-8 b(,)35 b Fn(Con)m(v)m(ergence)g(of)d (Probabilit)m(y)i(Measures)g Fx(\(Wiley)-8 b(,)34 b(New)g(Y)-8 b(ork,)32 b(1968\).)-24 3280 y([BS])50 b(E.)58 b(Bolthausen,)66 b(A.-S.)58 b(Sznitman,)66 b Fn(T)-8 b(en)59 b(lectures)h(on)e(random)g (media)p Fx(,)66 b(DMV)58 b(Seminar)h Fm(32)177 3401 y Fx(\(Birkh\177)-49 b(auser,)34 b(Basel,)f(2002\).)-24 3599 y([Bre])50 b(L.)32 b(Breiman,)i Fn(Probabilit)m(y)f Fx(\(Addison{W)-8 b(esley)g(,)36 b(Reading,)c(Massac)m(h)m(usetts,)k (1953\).)-24 3798 y([D)m(V])49 b(D.)31 b(J.)g(Daley)-8 b(,)31 b(D.)g(V)-8 b(ere{Jones,)32 b Fn(An)f(In)m(tro)s(duction)h(to)f (the)g(Theory)h(of)e(P)m(oin)m(t)i(Pro)s(cesses)h Fx(\(Springer,)177 3918 y(New)g(Y)-8 b(ork,)33 b(1988\).)-24 4117 y([DF)m(GW])49 b(A.)42 b(De)g(Masi,)j(P)-8 b(.)42 b(A.)h(F)-8 b(errari,)44 b(S.)e(Goldstein,)j(W.)e(D.)e(Wic)m(k,)46 b Fn(An)d(In)m(v)-5 b(ariance)43 b(Principle)h(for)177 4238 y(Rev)m(ersible)30 b(Mark)m(o)m(v)f(Pro)s(cesses.)g(Applications)g(to)e(Random)h(Motions)g (in)g(Random)f(En)m(vironmen)m(ts)p Fx(,)177 4358 y(J.)33 b(Stat.)f(Ph)m(ys.)j Fm(55)p Fx(,)d(787-855)f(\(1989\).)-24 4557 y([EF])49 b(A.)26 b(L.)f(Efros,)i(B.)e(I.)g(Shklo)m(vskii,)k Fn(Coulom)m(b)d(gap)f(and)g(lo)m(w)h(temp)s(erature)g(conductivit)m(y)h (of)e(disordered)177 4677 y(systems)p Fx(,)35 b(J.)e(Ph)m(ys.)h(C:)f (Solid)g(State)g(Ph)m(ys.)h Fm(8)p Fx(,)f(L49-L51)e(\(1975\).)-24 4876 y([FM])49 b(A.)42 b(F)-8 b(aggionato,)43 b(F.)e(Martinelli,)k Fn(Hydro)s(dynamic)e(limit)g(of)e(a)g(disordered)i(lattice)g(gas)p Fx(,)h(Probab.)177 4996 y(Theory)34 b(Related)f(Fields)g Fm(127)p Fx(,)g(535{608)e(\(2003\).)-24 5195 y([FKAS])49 b(P)-8 b(.)30 b(F)-8 b(rank)m(en,)31 b(D.)e(K\177)-49 b(onig,)30 b(U.)f(Arndt,)i(V.)f(Sc)m(hmidt,)h Fn(Queues)g(and)f(P)m (oin)m(t)g(Pro)s(cesses)i Fx(\(Ak)-5 b(adamie-)177 5315 y(V)d(erlag,)32 b(Berlin,)i(1981\).)1935 5564 y(46)p eop end %%Page: 47 47 TeXDict begin 47 46 bop -24 171 a Fx([Gri])49 b(G.)32 b(Grimmett,)h Fn(P)m(ercolation)p Fx(,)h(Second)f(Edition,)h (\(Springer)f(Grundlehren)g(321,)f(Berlin,)i(1999\))-24 374 y([KV])49 b(C.)42 b(Kipnis,)h(S.)e(R.)g(S.)g(V)-8 b(aradhan,)42 b Fn(Cen)m(tral)g(Limit)f(Theorem)h(for)e(Additiv)m(e)j (F)-8 b(unctionals)41 b(of)f(Re-)177 494 y(v)m(ersible)e(Mark)m(o)m(v)g (Pro)s(cesses)g(and)e(Applications)h(to)f(Simple)h(Exclusion)p Fx(,)i(Comm)m(un.)f(Math.)e(Ph)m(ys.)177 615 y Fm(104)p Fx(,)d(1-19)e(\(1986\).)-24 818 y([Kal])49 b(O.)c(Kallen)m(b)s(erg,)j Fn(F)-8 b(oundations)45 b(of)f(Mo)s(dern)i(Probabilit)m(y)p Fx(,)j(Second)c(Edition,)k(\(Springer-V)-8 b(erlag,)177 939 y(New)33 b(Y)-8 b(ork,)33 b(2001\).)-24 1142 y([KLP])50 b(W.)38 b(Kirsc)m(h,)i(O.)e(Lenoble,)i(L.)e(P)m(astur,)i Fn(On)e(the)h(Mott)f(form)m(ula)g(for)f(the)h(a.c.)h(conductivit)m(y)h (and)177 1262 y(binary)46 b(correlators)h(in)f(the)g(strong)g(lo)s (calization)g(regime)h(of)f(disordered)h(systems)p Fx(,)52 b(J.)46 b(Ph)m(ys.)i(A:)177 1383 y(Math.)33 b(Gen.)g Fm(36)p Fx(,)f(12157-12180)e(\(2003\))-24 1586 y([LB])49 b(F.)42 b(Ladieu,)j(J.-P)-8 b(.)42 b(Bouc)m(haud,)k Fn(Conductance)e (statistics)g(in)e(small)h(GaAs:Si)f(wires)i(at)d(lo)m(w)i(tem-)177 1706 y(p)s(eratures:)48 b(I.)35 b(Theoretical)h(analysis:)49 b(truncated)35 b(quan)m(tum)h(\015uctuations)g(in)f(insulating)g(wires) p Fx(,)i(J.)177 1827 y(Ph)m(ys.)d(I)f(F)-8 b(rance)33 b Fm(3)p Fx(,)f(2311-2320)f(\(1993\).)-24 2030 y([Mar])49 b(F.)37 b(Martinelli.)g Fs(L)-5 b(e)g(ctur)g(es)39 b(on)f(Glaub)-5 b(er)38 b(dynamics)g(for)g(discr)-5 b(ete)38 b(spin)g(mo)-5 b(dels)p Fx(.)35 b(Lecture)j(Notes)f(in)177 2151 y(Mathematics,)d(v)m (ol.)f(1717,)f(2000.)-24 2354 y([MKM])50 b(K.)32 b(Matthes,)h(J.)f (Kerstan,)h(J.)g(Mec)m(k)m(e,)h Fn(In\014nitely)g(Divisible)g(P)m(oin)m (t)f(Pro)s(cesses)h Fx(\(Wiley)f(series)h(in)177 2474 y(probabilit)m(y)f(and)g(mathematical)h(ph)m(ysics,)h(1978\))-24 2678 y([MR])50 b(R.)40 b(Meester,)j(R.)d(Ro)m(y)-8 b(,)43 b Fn(Con)m(tin)m(uum)f(P)m(ercolation)f Fx(\(Cam)m(bridge)g(Univ)m (ersit)m(y)i(Press,)h(Cam)m(bridge,)177 2798 y(1996\).)-24 3002 y([MA])50 b(A.)35 b(Miller,)i(E.)e(Abrahams,)h Fn(Impurit)m(y)h (Conduction)f(at)f(Lo)m(w)g(Concen)m(trations)p Fx(,)i(Ph)m(ys.)g(Rev.) f Fm(120)p Fx(,)177 3122 y(745-755)31 b(\(1960\))-24 3325 y([Min])50 b(N.)d(Minami,)52 b Fn(Lo)s(cal)47 b(\015uctuation)h 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y(ceedings)k(on)e(sto)s(c)m (hastic)h(metho)s(d)g(for)f(nonlinear)g(P)-8 b(.D.E.,)48 b(IMA)d(v)m(olumes)h(in)e(Mathematics)i Fm(77)p Fx(,)177 4658 y(Springer)33 b(V)-8 b(erlag,)33 b(New)g(Y)-8 b(ork,)33 b(65{79)e(\(1995\).)-24 4861 y([RS])49 b(M.)27 b(Reed,)i(B.)d(Simon,)j Fn(Metho)s(ds)e(of)f(Mo)s(dern)h(Mathematical)h(Ph)m(ysics)h(I-IV)e Fx(\(Academic)h(Press,)h(San)177 4982 y(Diego,)j(1980\).)-24 5185 y([Ros])49 b(M.)30 b(Rosen)m(blatt,)h Fn(Mark)m(o)m(v)g(Pro)s (cesses.)g(Structure)g(and)e(Asymptotic)i(Beha)m(vior)g Fx(\(Springer)f(Grund-)177 5305 y(lehren)k(184,)d(Berlin,)j(1971\).) 1935 5564 y(47)p eop end %%Page: 48 48 TeXDict begin 48 47 bop -24 171 a Fx([SE])50 b(B.)e(Shklo)m(vskii,)55 b(A.)49 b(L.)f(Efros,)k Fn(Electronic)e(Prop)s(erties)f(of)f(Dop)s(ed)g (Semiconductors)i Fx(\(Springer,)177 291 y(Berlin,)33 b(1984\).)-24 494 y([Sp)s(e])50 b(D.)24 b(Sp)s(ehner,)j Fn(Con)m(tributions)f(\023)-49 b(a)24 b(la)h(th)m(\023)-46 b(eorie)25 b(du)g(transp)s(ort)c(\023)-46 b(electronique)27 b(dissipatif)e(dans)g(les)h(solides)177 615 y(ap)m(\023)-46 b(erio)s(diques)p Fx(,)34 b(PhD)f(Thesis,)h(T)-8 b(oulouse,)34 b(2000.)-24 818 y([Sp)s(o])49 b(H.)33 b(Sp)s(ohn,)g Fn(Large)f(Scale)h (Dynamics)h(of)e(In)m(teracting)i(P)m(articles)p Fx(,)g(\(Springer,)f (Berlin,)h(1991\).)-24 1022 y([Tho])50 b(H.)33 b(Thorisson,)h Fn(Coupling,)f(Stationarit)m(y)-8 b(,)33 b(and)f(Regeneration)p Fx(,)i(\(Springer,)f(New)g(Y)-8 b(ork,)33 b(2000\).)1935 5564 y(48)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0407261047922--