Content-Type: multipart/mixed; boundary="-------------0011212355198" This is a multi-part message in MIME format. ---------------0011212355198 Content-Type: text/plain; name="00-463.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-463.keywords" random walk, pinning, wetting, limit theorem ---------------0011212355198 Content-Type: application/x-tex; name="rw" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="rw" % E. Bolthausen, H. Tanemura, N. Konno, M. 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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \bcenter \large{\bf One-sided random walk with weak pinning: pathwise descriptions of the phase transition \footnote{tentative title}} \normalsize \vvs Preliminary Draft \vvs Yasuki Isozaki\footnote{Osaka University email: yasuki@math.sci.osaka-u.ac.jp } and Nobuo Yoshida\footnote{Kyoto University email:nobuo@kusm.kyoto-u.ac.jp } \ecenter \begin{abstract} We consider a one-dimensional random walk which is conditioned to stay non-negative and is ``weakly pinned'' to zero. This model is known to exhibit a phase transition as the strength of the weak pinning varies. We prove path space limit theorems which describe the macroscopic shape of the path for all values of the pinning strength. If the pinning is less than (resp. equal to) the critical strength, then the limit process is the Brownian meander (resp. reflecting Brownian motion). If the pinning strength is supercritical, then the limit process is a positively recurrent Markov chain with a strong mixing property. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%% \SSC{Introduction} %%%%%%%%%%%%%%%%%%%%%%%%%%% In this paper, we discuss a localization-delocalization phase transition of a discrete interface above the attractive wall. Large number of works have been done on this subject, especially for one-dimensional interfaces, e.g. \cite{BDZ00,Bur81,Cha82,Fis84,Upt99,vLHi81} among all (See also \cite{BDZ00,CaVe00,DeVe00,DMRR92,IoVe00} for extensions in higher dimensions). The model considered in this paper is one-dimensional and is based on a symmetric, nearest neighbor random walk $(\{ S_n\}_{n \geq 0}, \{P^x\}_{x \in \Z})$ on $\Z$. To be precise, $\{ S_n\}_{n \geq 0}$ are integer valued random variables and $\{P^x\}_{x \in \Z}$ are probability measures such that the following hold for all $n \geq 1$ and $x \in \Z$; $S_1-S_0, \ldots, S_n-S_{n-1}$ are $P^x$-independent and \bdnl{Xn} P^x\{ S_0=x\}=1, \; \; \; P^x\{ S_n-S_{n-1}=0\}=r<1,\; \; \; P^x\{ S_n-S_{n-1}=\pm 1\}=v/2, \edn where $r+v=1$. For $n \geq 1$ and $x \in \Z$, we set \bdnl{L_n^x} L_n^x=\mbox{the number of $0 \leq j \leq n-1$ such that $S_j=x$}. \edn We define a probability measure $\m_n^x$ for $x,n \in \N=\{ 0,1, \ldots \}$ by $\m^x_0=P^x$ and \bdnl{Pn} \m^x_n (d\w)=\frac{1}{Z_n^x} P^x \lef[ \exp \lef( \b L_n^0\rig) : d\w \cap \{ S_1 \geq 0, \ldots, S_n \geq 0\} \ri], \; \; \; n \geq 1 \edn where $\b \in \R$ and $Z_n^x$ is the normalizing constant. In the case $\b>0$ which we are mainly interested in, the basic feature of the measure $\m^x_n$ can be explained as follows. We refer to the exponential factor in (\ref{Pn}) as the {\it weak pinning} which tries to localize the path $(S_k)^n_{k=1}$ near the origin. The effect of the weak pinning, however, has to compete with the entropic fluctuation of the random walk, which together with the restriction to $\{ S_1 \geq 0, \ldots, S_n \geq 0\} $, has a tendency to push the path away from the origin. We call the latter tendency {\it entropic repulsion}. The competetion between weak pinning and entropic repulsion has been well studied at the level of the free energy. We set \bdmn \b_c & = & \ln \frac{2}{1+r}, \label{b_c} \\ \lm_{\pm} & = & e^\b \frac{(e^\b-1)r\pm\sqrt{(e^\b-1)^2r^2 +(e^\b-1)v^2}}{2(e^\b-1)}, \; \; \; \mbox{for $\b >\b_c$.} \label{lmpm} \edmn Note that \bdnl{0<1<} \lm_{-}<0<1<\lm_{+}, \; \; \mbox{and}\; \; |\lm_{-}| < |\lm_{+}|. \edn It is known \cite{Bol99,Cha82,Fis84} and will also be seen later in this paper that the {\it free energy} for this model can explicitly be given as follows; \bdnl{frene} \psi (\b )\st{\rm def.}{=}\lim_{n \nearrow \8}n^{-1}\ln Z^x_n= \lef\{ \barray{ll} 0 & \mbox{if $\b \leq \b_c$, }\\ \ln (\lm_+)>0 & \mbox{if $\b >\b_c$.}\\ \earray \rig. \edn for all $x \in \N$. This means in particular that $\psi (\b)=\psi (0)$ for $\b \leq \b_c$ and $\psi (\b)>\psi (0)$ for $\b >\b_c$. We can therefore expect the following. For $\b<\b_c$, the effect of the weak pinning is smaller than that of the entropic repulsion and the pinning effect does not change the diffusive nature of the path in a qualitative way. On the other hand, if $\b >\b_c$, the effect of the weak pinning overwhelms that of the entropic repulsion and it confines the path near the origin. The purpose of this paper is to justify the above expected phenomena as limit theorems. Let $p \geq 0$, $m \geq 1$ and $f: I^m \ra \R$, where $I$ is either $[0,\8)$ or $\N$. We say $f$ is {\it $p$-bounded} if there is $p \geq 0$ such that \bdnl{polyb} \sup_{(x_j)^m_{j=1} \in I^m} (1+|x_1|+ \ldots +|x_m|)^{-p}|f(x_1, \ldots, x_m)|<\8. \edn We say $f$ is {\it polynomially bounded} if it is $p$-bounded for some $p\geq 0$. We introduce a bilinear form \bdnl{<>_b} \lan f_1,f_2 \ran_\b = e^{-\b} f_1(0)f_2(0)+ \sum_{x \geq 1 }f_1(x)f_2(x) \edn for $f_i: \N \ra \R$ ($i=1,2$) when the summation on the right-hand-side converges absolutely. For $\b >\b_c$, we define a function $\rh : \N \ra (0,1]$ by $\rh (x)=\rh (1)^x$, where \bdnl{rh} \rh (1)=\frac{-(e^\b-1)r+\sqrt{(e^\b-1)^2r^2 +(e^\b-1)v^2}} {(e^\b-1)v} \in (0,1). \edn %%%%%%%%%%%%%%%%%%% \Theorem{subsup} %%%%%%%%%%%%%%%%%% \bds \item[a)] If $\b \leq \b_c$, then \bdnl{subf} \limn \m_n^x\lef[ f(S_n/\sqrt{vn}) \rig] =\int_{[0,\8)}fd\m \edn for all $x \in \N$ and for any polynomially bounded continuous function $f:[0,\8) \ra \R$, where \bdnl{sublim} \m (dt)=\lef\{ \barray{ll} e^{-t^2/2}tdt & \mbox{if $\b <\b_c$, }\\ \sqrt{2/\pi }e^{-t^2/2}dt & \mbox{if $\b =\b_c$.}\\ \earray \rig. \edn \item[b)] For $\b > \b_c$, there are constants $C_{\ref{sup1}}=C_{\ref{sup1}}(\b, v)>0$ and $\e_{\ref{sup1}}=\e_{\ref{sup1}}(\b, v) \in (0,1)$ as follows; if $n \geq (1+x)C_{\ref{sup1}}$ and $f:\N \ra \R$ is polynomially bounded, then \bdnl{sup1} \lef| \m^x_n[f(S_n)]-\frac{\lan f, \rh \ran_\b }{\lan 1, \rh \ran_\b} \rig| \leq B_{\ref{sup1}}(f)\e_{\ref{sup1}}^{n}, \edn where $B_{\ref{sup1}}(f)$ is a constant which depends only on $\b$, $v$ and $f$. \eds %%%%%%%%%%%%%%% \end{theorem} %%%%%%%%%%%% We have the following process level limit theorem for $\b \leq \b_c$; %%%%%%%%%%%%%%%%%%% \Theorem{sub} %%%%%%%%%%%%%%%%%% For $\b \leq \b_c$, consider probability measures $\tl{\m}_n^x$, $n \geq 1$, $x \in \N$ on $C([0,1])$ defined by \bdnl{scale} \tl{\m}_n^x =\mu_n^x \lef\{ \lef( S_{nt}/\sqrt{v n} \rig)_{0 \leq t \leq 1} \in \cdot \rig\}, \edn where $(S_t)_{t \geq 0}$ is the linear interpolation of the random walk $(S_n)_{n \in \N}$. Then, for any fixed $x \in \N$, as $n \nearrow \8$, $\tl{\m}_n^x$ converges weakly to the law of a process $W=\lef( W_t\rig)_{0 \leq t \leq 1 }$. The process $W$ can be described as follows; \bdnl{W} W_t=\lef\{ \barray{ll} (1-\t )^{-1/2}B_{\t +t(1-\t ) } & \mbox{if $\b <\b_c$, }\\ |B_t| & \mbox{if $\b =\b_c$,}\\ \earray \rig. \edn where $B=\lef( B_t \rig)_{0 \leq t \leq 1 }$ is a one-dimensional Brownian motion starting from 0 and $\t =\sup\{ t \in [0,1] \; ; \; B_t=0 \}$. %%%%%%%%%%%%%%% \end{theorem} %%%%%%%%%%%% %%%%%%%%%%%%% \Remark{sub} %%%%%%%%%%%%% Similar limit theorems are discussed in the litratures. Bolthausen {\it et al} consider the case of one-dimensional weakly pinned Gaussian field in the delocalized phase \cite[Section4]{BDZ00}. In \cite{Upt99}, Upton discusses one-dimensional SOS interfaces and the phase boundaries of two-dimensional Ising model, where the thermodynamic paramerters are assumed to be sufficiently inside delocalized region. %%%%%%%%%% \end{remark} %%%%%%%%%%% We have a process level limit theorem also for $\b>\b_c$. We will have a positively recurrent Markov chain as the limit process. To describe the Markov chain obtaind as the limit, we introduce matrices $Q=(Q_{xy})_{x,y \in \N}$ and $R=(R_{xy})_{x,y \in \N}$ respectively by \bdn \label{Q} Q=\left( \begin{array}{ccccc} e^\b r & e^\b\frac{v}2 & 0 & 0 & \cdots\\ \frac{v}2 & r & \frac{v}2 & 0 & \cdots\\ 0 & \frac{v}2 & r & \frac{v}2 & \ddots\\ 0 & 0 & \frac{v}2 & r & \ddots \\ \vdots& \vdots& \ddots& \ddots& \ddots \end{array} \right) \edn and \bdn \label{R} R_{xy} = \lm_+^{-1}\rh (x)^{-1}Q_{xy}\rh (y). \edn We see by direct computation that \bdnl{psirh} \lm_+ =e^\b \lef( r+\frac{v}{2}\rh (1)\rig) = r+\frac{v}{2}\lef(\rh (1)+\rh (1)^{-1}\rig). \edn This implies that $\sum_{y \in \N }Q_{xy}\rh (y) =\lm_+ \rh (x) $ for all $x \in \N$ and hence that $R$ is a transition probability for an $\N$-valued Markov chain. We denote this Markov chain by $(\{S_n\}_{n \in \N}, \{ \nu^x \}_{x \in \N})$. It is easy to check that the Markov chain $\{ \nu^x \}_{x \in \N}$ is reversible with respect to the probability measure $ m $ on $\N$ defined by \bdnl{tlm(x)} m[f]=\frac{\lan f, \rh^2 \ran_\b }{\lan 1, \rh^2 \ran_\b} \; \; \; f: \N \ra [0,\8 ). \edn In particular, $\{ \nu^x \}_{x \in \N}$ is positively recurrent. %%%%%%%%%%%%%%%%%%% \Theorem{sup} %%%%%%%%%%%%%%%%%% For $\b>\b_c$, there are constants $C_{\ref{muranu}}=C_{\ref{muranu}}(\b,v)>0$ and $\e_{\ref{muranu}}=\e_{\ref{muranu}}(\b,v) \in (0,1)$ as follows; if $n \geq k +(1+x)C_{\ref{muranu}}$, $f: \N^k \ra \R $ is polynomially bounded and $F=f(S_1, \ldots, S_k)$, then \bdnl{muranu} |\mu^x_n [F]-\nu^x [F]| \leq B_{\ref{muranu}}(f) \e_{\ref{muranu}}^{n-k}, \edn where $B_{\ref{muranu}}(f)$ is a constant which depends only on $\b$, $v$ and $f$. %%%%%%%%%%%%%%% \end{theorem} %%%%%%%%%%%% By \Thm{sup}, we get the following strong mixing properties for both $\m^x_n$ and $\n^x$ for $\b>\b_c$; %%%%%%%%%%%%%%%%%%% \Theorem{decay} %%%%%%%%%%%%%%%%%% For $\b>\b_c$, there are constants $C_{\ref{decay}}=C_{\ref{decay}}(\b, v)>0$ and $\e_{\ref{decay}}=\e_{\ref{decay}}(\b, v) \in (0,1)$ as follows; if $n \geq l +m $, $l \geq k +(1+x)C_{\ref{decay}}$, $f: \N^k \ra \R $ and $g: \N^m \ra \R$ are polynomially bounded, $F=f(S_1, \ldots, S_k)$ and $G=g(S_{l+1}, \ldots, S_{l+m})$ then \bdmn |\mu_n^x (F\; G) -\mu_n^x (F)\mu_n^x (G)| & \leq & B_{\ref{decay}}(f,g) \e_{\ref{decay}}^{l-k}, \label{decay} \\ | \nu^x (F\; G) -\nu^x (F)\nu^x (G)| & \leq & B_{\ref{decay}}(f,g) \e_{\ref{decay}}^{l-k}, \label{nudec} \edmn where $B_{\ref{decay}}(f,g)$ is a constant which depends only on $\b$, $v$, $f$ and $g$. %%%%%%%%%%%%%%% \end{theorem} %%%%%%%%%%%% %%%%%%%%%% %%%%%%%%%%%%% \Remark{decay} %%%%%%%%%%%%% In \cite[Section4]{BDZ00}, it is mentioned that a similar statement to (\ref{decay}) is true for the one-dimensional weakly pinned Gaussian field in the localized phase. Localization of two-dimensional weakly pinned surfaces are discussed in \cite{DeVe00,DMRR92,IoVe00}, however without the attractive wall. %%%%%%%%%%%% \end{remark} %%%%%%%%%%%%% \SSC{Proof of \Thm{subsup} (a)} %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% \subsection{Preliminaries for the proof} %%%%%%%%%%%%%%%%%% For $x, n \in \N$, we introduce the following random variables; \bdmn h_n &= & 1\{ S_1 \geq 0, \ldots, S_n \geq 0\}, \; \; \; n \geq 1,\\ e_n & = & \exp \lef( \b L_n^0\rig), \; \; \; \b \in \R, n \geq 1, \\ z_n & = & e_nh_n, n \geq 1,\\ z_0 & = & e_0=h_0=1, \edmn where $1\{ \ldots \}$ denotes the indicator function. Then, the measure $\m^x_n$ can be rewritten as the following concise form; \bdnl{m^x_n} \m^x_n (d\w)=\frac{1}{Z_n^x}P^x[z_n: d\w], \; \; \; x,n \in \N, \edn We need to introduce some more notations. For a function $\vp :\N \ra \R$, a formal power series \bdnl{hatvp} \widehat{\vp}(s)=\sum_{n \in \N}s^n\vp (n), \; \; \; -1 \leq s \leq 1, \edn is called the {\it generating function} of $\vp$. If $\vp $ is non-negative and the convergence radius is one, then the following important relation between $\vp$ and $\widehat{\vp}$ is known as a Tauberian theorem \cite[page 447]{Fel70}; for $\a \in [0,\8 )$ and $C>0$, \bdnl{Tauber} \widehat{\vp}(s) \st{s \nearrow 1}{\sim}C(1-s)^{-\a} \; \; \; \mbox{if and only if} \; \; \; \vp (0)+\cdots +\vp (n-1) \st{n \nearrow \8 }{\sim}\frac{Cn^\a}{\Gm (\a +1)}. \edn Moreover, if $\a \neq 0$ and $\vp$ is monotone, then (\ref{Tauber}) is equivalent to \bdnl{mTauber} \vp (n) \st{n \nearrow \8 }{\sim}\frac{Cn^{\a-1}}{\Gm (\a )}. \edn We set \bdnl{f^x} f^x(n)= \lef\{ \barray{ll} 0 & \mbox{if $n =0$, }\\ P^x\{ T_1=n\}, & \mbox{if $n \geq 1$,}\\ \earray \rig. \edn where $T_m$ denotes the $m^{\rm th}$-hitting time to the origin; \bdnl{T_m} T_m=\inf \lef\{ n \geq 1 \; | \; \sum_{k=1}^n1\{ S_k =0\} =m \rig\}. \edn It is then well known that the generating function is given by \bdnl{^f^x} \widehat{f^x}(s) =\lef\{ \barray{ll} \lef( \frac{1-rs-\gm (s)^{1/2}}{(1-r)s}\rig)^{|x|} & \mbox{if $x \neq 0$, $|s|<1$, }\\ 1-\gm (s)^{1/2} & \mbox{if $x = 0$, $|s|<1$,}\\ \earray \rig. \edn where $\gm (s)=(1-rs)^2-s^2v^2$. %%%%%%%%%%%%%%%%%% \subsection{Lemmas} %%%%%%%%%%%%%%%%%% We define \bdn \tht_{\pm} (s) = 1-\frac{e^\b}{2} \lef( rs+1\pm \sqrt{\gm (s)} \rig) \; \; \; \mbox{for $s \in (-\8,1]$}. \label{tht(s)} \edn It is then easy to see that \bdnl{thzero} \{ s \in (-\8,1]\; ; \; \tht_{-} (s)=0\}= \lef\{ \barray{ll} \epty & \mbox{if $\b <\b_c$, }\\ \{ 1 \} & \mbox{if $\b =\b_c$, }\\ \{ 1/\lm_-, 1/\lm_+\} & \mbox{if $\b >\b_c$,} \earray \rig. \edn and that $\tht_{-} (s) $ is strictly decreasing on $[0,1]$. %%%%%%%%%%%%%%%%%%%% \Lemma{^Z^xy} %%%%%%%%%%%%%%%%%%%%% For $x,y,n \in \N$, define \bdmn z_n^{y} & = & z_n1\{ S_n=y \}, \label{z^xy} \\ Z_n^{x,y} & = & P^x[z_n^{y}]. \label{Z^xy} \edmn Then, for $0 \leq s <1$, \bdnl{^Z^x0} \widehat{Z}^{x,0}(s)=\frac{1}{\tht_{-} (s) \vee 0} \lef( \del^{x,0}+(1-\del^{x,0})\widehat{f^x}(s)\rig). \edn In particular, \bdn \widehat{Z}^{x,0}(1) = 1/\tht_{-} (1) \; \; \; \mbox{if $\b <\b_c$, } \label{^Z^x01} \edn and \bdnl{^Z^x0c} \widehat{Z}^{x,0}(s) \st{s \nearrow 1}{\sim} C_{\ref{^Z^x0c}}/(1-s)^{1/2}, \; \; \; \mbox{if $\b = \b_c$. }\\ \edn where $C_{\ref{^Z^x0c}}=\frac{1+r}{\sqrt{2v}}$. %%%%%%%%%%%%%%% \end{lemma} %%%%%%%%%%%%%%% Proof: The equality (\ref{^Z^x0}) is essentially the same as the representation of Green function of the random walk in terms of its hitting times. We follow \cite{Bol99}. Note first that \bdnl{0n1} \widehat{Z}^{x,0}(s)=\del^{x,0}+\sum_{n \geq 1}s^nP^x[z^0_n]. \edn To compute the right-hand-side, let us introduce \bdnl{g^x} g^x(n) =\lef\{ \barray{ll} 0 & \mbox{if $n =0$, }\\ P^x\{h_n : T_1=n\} & \mbox{if $n \geq 1$.}\\ \earray \rig. \edn It is then, easy to see that \bdnl{g^xf^x} g^x (n)=\lef\{ \barray{ll} f^x (n) & \mbox{if $x \geq 1$, }\\ f^x (n)=r & \mbox{if $x =0$ and n=1, }\\ f^x (n)/2 & \mbox{if $x =0$ and $n \geq 2$. }\\ \earray \rig. \edn and consequently that for $0 \leq s \leq 1$, \bdnl{^g^x} \widehat{g^x}(s) =\lef\{ \barray{ll} (rs +\widehat{f^0}(s))/2 & \mbox{if $x =0$, }\\ \widehat{f^x}(s) & \mbox{if $x \geq 1$. } \earray \rig. \edn We compute the right-hand-side of (\ref{0n1}), first for the case $x \geq 1$; \bdmn \lefteqn{\sum_{n \geq 1}s^nP^x[z^0_n] } \\ & = & \sum_{n \geq 1}s^n\sum_{m \geq 1}e^{(m-1)\b} \sum_{0< i_1 < \ldots n \}$. To prove this, we set \bdnn k_{m,n} & = & 1\{ S_m >0, \ldots, S_n >0\}, \; \; \; 1 \leq m \leq n,\\ d_n & = & \max D_n \; \; \; \mbox{ if $D_n\st{\rm def.}{=} \{ 0 \leq j \leq n \; ; \; S_j=0 \} \neq \epty$.} \ednn We have \bdnl{P^xk_n} P^x[ k_{1,n}] =\lef\{ \barray{ll} F^x_n/2& \mbox{if $x =0$, }\\ F^x_n & \mbox{if $x \geq 1$. } \earray \rig. \edn Note also that $D_n =\epty$ if and only if $x \geq 1$ and $T_1 >n$. Since we have either $d_n \in \{ 0, \ldots ,n\}$ or $D_n =\epty$, we see that $z_n$ can be decomposed as follows; \bdnn z_n & = & \sum_{m=0}^{n-1}z_n1\{ d_n=m \}+z_n1\{ d_n=n\} +(1-\del^{0,x})z_n1\{ T_1 >n\} \\ & = & \sum_{m=0}^{n-1}e^\b z_m^0k_{m+1,n} +z_n^0+(1-\del^{0,x})k_{1,n}. \ednn We therefore have \bdnn Z^x_n & = & e^\b\sum_{m=0}^{n-1}P^x[z_m^0k_{m+1,n}] +Z^{x,0}_n+(1-\del^{x,0})P^x[ k_{1,n}] \\ & = & \half e^\b \sum_{m=0}^{n-1}Z^{x,0}_mF^0_{n-m}+Z^{x,0}_n +(1-\del^{x,0})F^x_n, \ednn where we have used (\ref{P^xk_n}) in the last line. This proves (\ref{ZZ}). It is easy to see that \bdnl{^F} \widehat{F^x}(s)=\frac{1-\widehat{f^x}(s)}{1-s}. \edn Therefore, (\ref{^Z^x}) is an easy consequence of (\ref{^Z^x0}), (\ref{^F}) and (\ref{ZZ}). We see (\ref{^Z^x1}) from (\ref{^Z^x}) and the following asymptotic property for $\widehat{f^x}(s)$; \bdnl{sim^f} 1-\widehat{f^x}(s) \sim \lef\{ \barray{ll} \sqrt{2v(1-s)}& \mbox{if $x=0$, }\\ x\sqrt{2(1-s)/v}& \mbox{if $x \neq 0$, }\\ \earray \rig. \; \; \; \mbox{as $s \nearrow 1$.} \edn %%%%%%%%%%%% $\Box$ \vs %%%%%%%%%%%% To state and prove the next lemma, we will use the following notation. For a sequence $(a_n )_{n \geq 0}$ (random or non-random), we set $\D a_n=a_n-a_{n-1}$ for $n \geq 1$. %%%%%%%%%%%%%%%%%%%% \Lemma{ZS^p} %%%%%%%%%%%%%%%%%%%%% Let $\Phi_0$ be a constant and $\Phi_n=\vp_n(S_1, \ldots, S_n)$ for $n \geq 1$, where each $\vp_n$ is a function on $\N^n$. Then, \bdmn P^x[\Phi_nz_n] & = & \Phi_0 +\sum_{k=1}^nP^x[z_{k-1}\D \Phi_k ] +\sum_{k=1}^nP^x[z_{k-1}^0 (e^\b1\{ \D S_k \geq 0 \}-1)\D \Phi_k ] \nn \\ & & -\tht_{-} (1)\sum_{k=1}^nP^x[z_{k-1}^0\Phi_{k-1}]. \label{PFz} \edmn In particular, \bdmn Z^x_n & = & 1-\tht_{-} (1)\sum_{k=0}^{n-1}Z^{x,0}_k, \label{p=0} \\ P^x[S^p_nz_n] & = & x^p +v\sum_{1 \leq j \leq p/2}\frac{p!}{(2j)!(p-2j)!} \sum_{k=0}^{n-1}P^x[S^{p-2j}_kz_k] \nn \\ & & +\frac{v}{2}\{ e^\b -1-(-1)^p \} \sum_{k=0}^{n-1}Z^{x,0}_k. \label{p>0} \edmn %%%%%%%%%%%%%%% \end{lemma} %%%%%%%%%%%%%%% Proof: %%%%%%%%% We observe that $$ \D z_n =\lef( e^\b1\{ \D S_n \geq 0\}-1 \rig)z_{n-1}^0. $$ We therefore have \bdnn \D (z_n\Phi_n) & = & z_n\D \Phi_n+\Phi_{n-1}\D z_n \\ & = & (z_{n-1}+\D z_n)\D \Phi_n+\Phi_{n-1}\D z_n \\ & = & z_{n-1}\D \Phi_n+\lef( e^\b1\{ \D S_n \geq 0\}-1 \rig)z_{n-1}^0\D \Phi_n +\lef( e^\b1\{ \D S_n \geq 0\}-1 \rig)z_{n-1}^0\Phi_{n-1} \ednn This implies (\ref{PFz}), since $P^x[\Phi_nz_n] = \Phi_0+\sum_{k=1}^nP^x[\D (z_k \Phi_k )] $ and $P^x[ e^\b1\{ \D S_n \geq 0 \}-1]=-\tht_{-} (1)$. The other two equalities are obtained easily from (\ref{PFz}). %%%%%%%%%%%% $\Box$ \vs %%%%%%%%%%%% \subsection{Proof of \Thm{subsup}({\bf a})} %%%%%%%% It is enough to prove that for all $x,p \in \N$, \bdnl{sub1} \limn \m_n^x\lef[ (S_n/\sqrt{vn})^p \rig]= \int^\8_0x^p\m (dx) =\lef\{ \barray{ll} 2^{p/2}\Gm (\mbox{$\frac{p}{2}$}+1) & \mbox{if $\b <\b_c$, }\\ \sqrt{2^p/\pi }\Gm (\frac{p+1}{2}) & \mbox{if $\b =\b_c$.}\\ \earray \rig. \edn We begin with the case $\b <\b_c$. Since $Z_n^x$ is decreasing in $n$ by (\ref{p=0}), we see from (\ref{^Z^x1}) and (\ref{mTauber}) that \bdnl{Znsim} Z_n^x{\sim}C_{\ref{^Z^x1}}/(\pi n)^{1/2} \; \; \; \mbox{as $n \nearrow \8$.} \edn Therefore, if we set $M_n(p)=P^x[S_n^pz_n]$, then, (\ref{sub1}) for $\b <\b_c$ is equivalent to that; \bdnl{esub1} \limn n^{-\frac{p-1}{2}}M_n(p) = (C_{\ref{^Z^x1}}/\sqrt{\pi})(2v)^{p/2}\Gm (\mbox{$\frac{p}{2}$}+1). \edn We prove (\ref{esub1}) by induction. For $p=0$, (\ref{esub1}) can immediately be seen from (\ref{Znsim}). For $p=1$, (\ref{esub1}) can be seen from (\ref{^Z^x01}), (\ref{p>0}) and (\ref{Znsim}). Let us consider the case $p \geq 2$. We may now assume (\ref{esub1}) with $p$ replaced by $p-2j$ ($1 \leq j \leq p/2$). We denote the right-hand-side of (\ref{esub1}) by $m(p)$. It is then not difficult to see that \bdmn \lefteqn{n^{-\frac{p-1}{2}}\sum_{k=0}^{n-1}M_k(p-2j)} \nn \\ & = & n^{-\frac{p-1}{2}}\sum_{k=0}^{n-1} k^{-\frac{p-2j-1}{2}} M_k(p-2j)k^{\frac{p-2j-1}{2}} \st{n \nearrow \8}{\lra} \lef\{ \barray{ll} \frac{m(p-2)}{(p-1)/2}, & \mbox{if $j=1$, }\\ 0 & \mbox{if otherwise.}\\ \earray \rig. \label{limp-1} \edmn We have $$ M_n(p)= v\sum_{1 \leq j \leq p/2}\frac{p!}{(2j)!(p-2j)!} \sum_{k=0}^{n-1}M_k(p-2j)+\cO (1) \; \; \; \mbox{as $n \nearrow \8$} $$ by (\ref{^Z^x01}) and (\ref{p>0}). We see from this and (\ref{limp-1}) that \bdnn \limn n^{-\frac{p-1}{2}}M_n(p) & = & v\frac{p!}{2!(p-2)!}\frac{m(p-2)}{(p-1)/2} \\ & = & vpm(p-2)=m(p), \ednn which completes the proof of (\ref{esub1}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Proof of (\ref{sub1}) for $\b=\b_c$ is similar as above. What we have to show in this case is that; \bdnl{esub1c} \limn n^{-\frac{p}{2}}M_n(p) =\sqrt{(2v)^p/\pi }\Gm (\mbox{$\frac{p+1}{2}$}). \edn We prove (\ref{esub1c}) by induction. For $p=0$, (\ref{esub1c}) can immediately be seen from (\ref{p=0}). For $p=1$, we see from (\ref{^Z^x0c}) and (\ref{Tauber}) that \bdnl{Znsimc} \sum_{k=0}^{n-1}Z^{x,0}_k {\sim}2C_{\ref{^Z^x0c}}n^{1/2}/\pi \; \; \; \mbox{as $n \nearrow \8$.} \edn This and (\ref{p>0}) imply (\ref{esub1c}) for $p=1$. Let us consider the case $p \geq 2$. We may now assume (\ref{esub1c}) with $p$ replaced by $p-2j$ ($1 \leq j \leq p/2$). We denote the right-hand-side of (\ref{esub1c}) by $m(p)$. It is then not difficult to see that \bdmn \lefteqn{n^{-\frac{p}{2}}\sum_{k=0}^{n-1}M_k(p-2j)} \nn \\ & = & n^{-\frac{p}{2}}\sum_{k=0}^{n-1} k^{-\frac{p-2j}{2}} M_k(p-2j)k^{\frac{p-2j}{2}} \st{n \nearrow \8}{\lra} \lef\{ \barray{ll} \frac{m(p-2)}{p/2} & \mbox{if $j=1$, }\\ 0 & \mbox{if otherwise.}\\ \earray \rig. \label{limp-c} \edmn We have $$ M_n(p)= v\sum_{1 \leq j \leq p/2}\frac{p!}{(2j)!(p-2j)!} \sum_{k=0}^{n-1}M_k(p-2j)+\cO (n^{1/2}) \; \; \; \mbox{as $n \nearrow \8$} $$ by (\ref{p>0}) and (\ref{Znsimc}). We see from this and (\ref{limp-c}) that \bdnn \limn n^{-\frac{p}{2}}M_n(p) & = & v\frac{p!}{2!(p-2)!}\frac{m(p-2)}{p/2}\\ & = & v(p-1)m(p-2)=m(p), \ednn which completes the proof of (\ref{esub1c}). %%%%%%% $\Box$ %%%%%%%%%%% \SSC{Proof of \Thm{sub}} %%%%%%%%%% We consider the case $x=0$ for simplicity. The modifications if otherwise are straightforward. %%%%%%%%%%%%%% \subsection{The critical case} %%%%%%%%%%% In this case, the paths of $\{ S_n \}$ can be embedded in the paths of $W(t)\equiv |B(t)|$ without changing the law under $\mu_n$. Note first that for all $k0,&& \mu_n\{ \D S_k=0| S_{k-1}\}=r; \mu_n\{ \D S_k= \pm1 | S_{k-1}\}=v/2. \label{k-1>0}\\ \mbox{If } S_{k-1}=0,&& \mu_n\{ \D S_k=0 | S_{k-1}\}=\frac{2r}{1+r}; \mu_n\{ \D S_k= 1 | S_{k-1}\}=\frac{v}{1+r}. \label{k-1=0} \edmn By modifying the Skhrohod embedding, we can construct a series of stopping times $\{ \s_k \}_{k \geq 0}$ so that $\{ W(\s_k) \}_{k=1}^n$ is a realization of $(\{ S_k\}_{k=1}^n, \m^0_n)$. In fact, for constants $K_0$ and $K_1$ we define $\{ \s_k \}_{k \geq 0}$ as follows; $\s_0:=0$ and if $W(\s_{k-1})\in \N\setminus\{0\}$ then $$ \s_k:= \inf\{ t \geq \s_{k-1}; |W(t)-W(\s_{k-1})|=1 \mbox{ or } (t>K_1+\s_{k-1} \mbox{ and } W(t)=W(\s_{k-1})) \} $$ if $W(\s_{k-1})=0$ then $$ \s_k:= \inf\{ t \geq \s_{k-1}; W(t)=1 \mbox{ or } (t>K_0+\s_{k-1} \mbox{ and } W(t)=0 ) \}. $$ We can then, choose $K_0$ and $K_1$ (depending upon $\b$ and $v$) so that $\tl{S_k}\st{\rm def.}{=}W(\s_k)$ satisfies \bdmn \mbox{If } \tl{S}_{k-1}>0,&& P^W\{ \D \tl{S}_k=0| \tl{S}_{k-1}\}=r; P^W\{ \D \tl{S}_k= \pm1 | \tl{S}_{k-1}\}=v/2. \label{tlk-1>0}\\ \mbox{If } \tl{S}_{k-1}=0,&& P^W\{ \D \tl{S}_k=0 | \tl{S}_{k-1}\}=\frac{2r}{1+r}; P^W\{ \D \tl{S}_k= 1 | \tl{S}_{k-1}\}=\frac{v}{1+r}. \label{tlk-1=0} \edmn By discarding $k$'s such that $W(\s_{k-1})=0$, we obtain a series of i.i.d. $\{ \t_k ; {k \geq 0}\} := \{ \s_k-\s_{k-1} ; {k \geq 0}, W(\s_{k-1})\neq0\}$. It is elementary to prove $E^W[ \t_k]=v$. By well-known Kolmogorov's inequality, we have \bdn \label{ByKolmIneq} P^W[ \exists k\leq n, \; |\sum_{j=1}^k (\t_j-v)|>\delta n ]\leq Var(\t_1)/\delta^2n. \edn Since the number of downcrossings from $1/\sqrt{n}$ to $0$ approximate the local time, it holds, after rescaled by $\sqrt{n}$, for any $\e>0$, there is $K_2>0$ such that for all large $n$, \bdn \label{Visit-0-Mean} P^W[ \#\{ k\leq n; W(\s_{k-1})=0\}>K_2 \sqrt{n}/v]<\e . \edn The following is its natural consequence: there is $K_3>0$ such that for all large $n$, \bdn \label{Visit-0-RV} P^W[ \sum_{ k\leq n; W(\s_{k-1})=0} (\s_{k}-\s_{k-1}) >K_3 \sqrt{n}]<\e . \edn Choose $n$ so large that $1/n<\delta$ and $(K_2\vee K_3)\sqrt{n}<\delta n$, where $\delta$ if beforehand chosen so that \bdn \label{Conti-W} P^W[ \sup_{0\e \sqrt{n}]<\e . \edn Now it remains to put these together: \bdnn \sup_{00\; ; \; 1 \leq j \leq k, \; n \geq 1\}$ are such that $00, \; x \in \R,\\ h(x) & = & 2\int^x_0g_1 (y)dy, \; \; \; x \geq 0,\\ g_t (x,y) & = & g_t (x-y)-g_t (x+y), \; \; \; t>0, \; x,y \in \R,\\ h(t,x) & = & h\lef( \frac{x}{\sqrt{1-t}}\rig), \; \; \; (t,x) \in [0,1] \times [0,\8 ) \bsh \{ (1,0)\} \edmn According to Belkin \cite{Bel72}, the process $W$ for $\b <\b_c$ is a (time-inhomogeneous) diffusion process on $[0,\8)$ with the following transition probability; for $0 \leq s 0$ and $x >0$,}\\ 0 & \mbox{if otherwise} \earray \rig. \edn We set \bdnl{S^minmax} S^{\min }_n=\min_{0 \leq j \leq n}S_j. %\; \; \; \mbox{and}\; \; \; %S^{\max }_n=\max_{0 \leq j \leq n}S_j. \edn %%%%%%%%%%%%%%%%%%%% \Lemma{hp} %%%%%%%%%%%%%%%%%%%%% Let $\b <\b_c$. Then, for $0 \leq s 0$. In particular, \bdn \limn Z^{y\sqrt{vn}}_{n-\lfl ns \rfl} = h(s,y). \label{limZ=h} \edn uniformly in $y \in [0,b]$ for any $b>0$. %%%%%%%%%%%%%%% \end{lemma} %%%%%%%%%%%%%%% Proof: %%%%%%%%%%%% We write $\lan u \ran =\lfl u\rfl-\lfl ns\rfl$ ($u \geq 0$) in this proof. We define $$ \t =\t (y, n)=\inf \{ j \geq 1\; ; \; S_j \leq S_0-y\sqrt{vn} \} $$ and divide the expectation on the left-hand-side of (\ref{limhp}) as follows; \bdmn \lefteqn{P^{y\sqrt{vn}}\lef[ z_{\lan n \ran }: S_{\lan nt \ran } \leq x\sqrt{vn}\ri] } \nn \\ & = & P^0\lef[ \exp \lef( \b L_{\lan n \ran}^{-y\sqrt{vn}}\ri): \frac{S^{\min }_{ \lan n \ran}}{\sqrt{vn}} \geq -y, \; \frac{S_{\lan nt \ran }}{\sqrt{vn}} \leq x-y\ri] \nn \\ & = & P^0\lef[ \frac{S^{\min }_{ \lan n \ran}}{\sqrt{vn}} > -y, \; \frac{S_{\lan nt \ran}}{\sqrt{vn}} \leq x-y\ri] \label{1sthp}\\ & & +\sum^{\lan n \ran -1}_{k=0} P^0\lef[ \exp \lef( \b L_{\lan n \ran}^{-y\sqrt{vn}}\ri): \frac{S^{\min }_{ \lan n \ran}}{\sqrt{vn}} \geq -y, \; \frac{S_{\lan nt \ran}}{\sqrt{vn}} \leq x-y, \; \t=k \ri] \label{2ndhp} \edmn The first term (\ref{1sthp}) converges to the right-hand-side of (\ref{limhp}) by the invariance principle of Donsker. Moreover, the convergence is uniform in $(x,y) \in [0,\8)^2$ by a result of Billingsley and Tops$\phi$e \cite[Theorem 2]{BiTo67}. On the other hand, the second term (\ref{2ndhp}) is bounded from above by \bdnl{3rddhp} \sum^{\lan n \ran -1}_{k=0} P^0\{ \t =k \}Z^0_{\lan n \ran -k}. \edn By using (\ref{Znsim}) and the fact that $P^0\{ \t =k \}$ is not greater than a constant times $y\sqrt{n}k^{-3/2}$, it is not difficult to bound (\ref{3rddhp}) by a constant times $yn^{-1/3}$. This completes the proof. %%%%%%%%%%% $\Box$ %%%%%%%%% \vs Proof of \Lem{fdd}: %%%%%%%%%% Note first that it is enough to prove (\ref{fddvect}) for a special case: $t_{j,n}=\lfl nt_j \rfl$ ($1 \leq j \leq k$). To see this, let $\{ t_{j,n} \}$ be an arbitrary sequence which satisfies the assumptions in \Lem{fdd}. Since $S_{t_{j,n}}$ and $S_{\lfl nt_j \rfl}$ differ at most by $|t_{j,n}-\lfl nt_j \rfl|$, the $L^1(\mu^0_n)$-difference of two vectors $\lef(S_{t_{j,n}}/\sqrt{vn} \rig)^k_{j=1}$ and $\lef(S_{\lfl nt_j \rfl}/\sqrt{vn} \rig)^k_{j=1}$ vanishes as $n \nearrow \8$. Therefore, their limits in law coincide. Following Iglehart \cite[proof of (2.23)]{Igl74}, we prove (\ref{fddvect}) by induction on $k$. We will prove that for all $x_1, \ldots, x_k >0$, \bdn \limn \m_n^0 (A_k) = \int^{x_1}_0dy_1 \ldots \int^{x_k}_0dy_k p(0,0|t_1,y_1)p(t_1,y_1|t_2,y_2)\cdots p(t_{k-1},y_{k-1}|t_k,y_k), \label{cfdd} \edn where $A_k=\lef\{ \frac{S_{\lfl nt_1 \rfl}}{\sqrt{vn}} \leq x_1, \ldots ,\frac{S_{\lfl nt_k \rfl}}{\sqrt{vn}} \leq x_k \rig\}$. We set $(s,t,x)=(t_{k-1},t_k,x_k)$ to simplify the expressions. We start with the case $k=1$. If $t=1$, then, (\ref{cfdd}) follows from (\ref{subf}). Let us therefore assume $t<1$. We then have \bdnl{k=1} \m^0_n \lef\{ \frac{S_{\lfl nt \rfl}}{\sqrt{vn}} \leq x \ri\} =\frac{Z^0_{\lfl nt \rfl}}{Z^0_n}\int_{y \leq x} \m^0_{\lfl nt \rfl} \lef\{ \frac{S_{\lfl nt \rfl}}{\sqrt{vn}} \in dy \ri\} Z^{y\sqrt{vn}}_{n-\lfl nt \rfl} \edn We see from \Lem{hp} that $Z^{y\sqrt{vn}}_{n-\lfl nt \rfl}$ converges to $h\lef(t, y\ri)$ locally uniformly in $y$. We therefore have by (\ref{subf}) and (\ref{Znsim}) that \bdmn \limn \m^0_n \lef\{ \frac{S_{\lfl nt \rfl}}{\sqrt{vn}} \leq x \ri\} & = & t^{-1/2}\int^{x}_0\m (t^{-1/2}dy ) h\lef( t,y\ri) dy \nn \\ & = & \int^{x}_0p(0,0| t,y)dy \nn \edmn which proves (\ref{cfdd}) for $k=1$. We now consider the case $k \geq 2$. Since $$ \frac{S_{\lfl nt_j \rfl}}{\sqrt{vn}}= \sqrt{\frac{m}{n}} \frac{S_{m \cdot \lfl nt_j \rfl /m}}{\sqrt{vm}}, \; \; \; m=\lfl ns \rfl, $$ we have by the induction hypothesis and the observation at the beginning of the proof that $$ \limn \m_{\lfl ns \rfl}^0 \lef\{ \lef(S_{\lfl nt_j \rfl}/\sqrt{vn} \rig)^{k-1}_{j=1} \in \cdot \rig\} =\mbox{the law of $\lef(\sqrt{s}W_{t_j/s} \rig)^{k-1}_{j=1}$.} $$ Note that the above limit has the density \bdnl{fddk-1} p_s\lef( 0,0 | t_1,y_1 \ri) p_s\lef( t_1,y_1| t_2,y_2 \ri) \cdots p_s\lef( t_{k-2},y_{k-2}| s,y_{k-1} \ri), \edn where $$ p_s\lef( t,x | t^\prime,y \ri)= s^{-1/2}p\lef( t/s,x/\sqrt{s} | t^\prime/s,y/\sqrt{s} \ri). $$ If we set $h_s (t,x)=h(t/s,x/\sqrt{s})$, then we have obviously that \bdnl{p and p_s} p_s\lef( t,x | t^\prime,y \ri)= \lef\{ \barray{ll} s^{1/2}p\lef( 0,0 | t^\prime/s,y/\sqrt{s} \ri) \frac{h_s (t^\prime,y)}{h (t^\prime,y)}, & \mbox{if $t=0$ and $x=0$}\\ \frac{h_s (t,x)}{h (t,x)} p\lef( t/s,x/\sqrt{s} | t^\prime/s,y/\sqrt{s} \ri) \frac{h_s (t^\prime,y)}{h (t^\prime,y)}, & \mbox{if $t>0$ and $x >0$.}\\ \earray \rig. \edn We rewrite the probability on the left-hand-side of (\ref{cfdd}) as follows; \bdnl{Ak-1} \m_n^0 (A_k) =\frac{Z^0_{\lfl ns \rfl}}{Z^0_n}\int^{x_{k-1}}_0 \m^0_{\lfl ns \rfl}\lef\{ A_{k-2}, \frac{S_{\lfl ns \rfl}}{\sqrt{vn}} \in dy \ri\} P^{y\sqrt{vn}}\lef[ z_{n-\lfl ns \rfl}: \frac{S_{\lfl nt \rfl-\lfl ns \rfl}}{\sqrt{vn}} \leq x\ri] \edn By (\ref{Znsim}), (\ref{Ak-1}), (\ref{limhp}) and (\ref{p and p_s}) we conclude that \bdmn \lefteqn{\limn \m_n^0 (A_k) } \nn \\ & = & s^{-1/2}\int^{x_1}_0dy_1 \ldots \int^{x_{k-2}}_0dy_{k-2}\int^{x_{k-1}}_0dy p_s\lef( 0,0 | t_1,y_1 \ri) p_s\lef( t_1,y_1| t_2,y_2 \ri) \cdots p_s\lef( t_{k-2},y_{k-2}| s,y \ri) \nn \\ & & \times h(s,y)\int^x_0p(s,y |t,z)dz \nn \\ & = & \int^{x_1}_0dy_1 \ldots \int^{x_k}_0dy_k p(0,0|t_1,y_1)p(t_1,y_1|t_2,y_2)\cdots p(t_{k-1},y_{k-1}|t_k,y_k). \edmn This completes the proof of \Lem{fdd}. %%%%%%%%%%% $\Box$ %%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Tightness for ${\bf \b <\b_c}$} %%%%%%%%%%%%%%%%%%%% We continue to consider the distribution of $(S_{nt}/\sqrt{vn})_{0 \leq t \leq 1}$ under $\m^0_n $ with $\b <\b_c$. We have shown in \Lem{fdd} that any finite dimensional distribution of converges. The purpose of this subsection is to prove the tightness of the distribution, which, together with \Lem{fdd}, implies \Thm{sub} for $\b <\b_c$. By the well-known Kolmogorov's tightness criterion, it is enough to prove that for $\e>0$, \bdnl{tight} \lim_{\del \searrow 0}\sup_{n \geq 1}\m^0_n\lef\{ \max_{\barray{l} \mbox{\scriptsize $t,u \leq 1$} \\ \mbox{\scriptsize $|t-u| \leq \del $} \earray}|S_{nt}-S_{nu}| \geq \e \sqrt{n}\ri\}=0, \edn It is easy to see that (\ref{tight}) follows from the following estimate. %%%%%%%%%%%%%%%%%%%% \Lemma{A_{n,l}} %%%%%%%%%%%%%%%%%%%%% For $\b <\b_c$, there is a constant $C_{\ref{Mk+l,k}}=C_{\ref{Mk+l,k}}(\b,v) \in (0,\8)$ such that for all $l \geq 1$ and $\lm \geq 1$; \bdnl{Mk+l,k} \m_n^0\lef\{ \max_{\barray{l} \mbox{\scriptsize $0 \leq j \leq l$} \\ \mbox{\scriptsize $0 \leq k \leq n-j$} \earray}|S_{k+j}-S_{k}| \geq \lm \sqrt{vl} \ri\} \leq C_{\ref{Mk+l,k}}\lef( n/l\ri)^{3/2} \exp \lef( -\lm /C_{\ref{Mk+l,k}} \ri). \edn %%%%%%%%%%%%%%% \end{lemma} %%%%%%%%%%%%%%% We first prove the following exponential integrability property. %%%%%%%%%%%%%%% \Lemma{expint} %%%%%%%%%%%% Let $\b <\b_c$ and $\e>0$ be such that \bdnl{c-expint} \frac{v}{2}(e^\e -1)(e^\b -1) \leq \tht_{-}(1). \edn Then, \bdnl{r-expint} \sup_{n \geq \e^2}P^x\lef[ z_n \exp \lef( \e\lef| \frac{S_n-x}{\sqrt{n}}\ri| \ri)\ri] \leq 2 \exp \lef( \frac{\cosh (1)v\e^2}{2} \ri). \edn %%%%%%%%%%%%%%% \end{lemma} %%%%%%%%%%%%%%% Proof: We will show that \bdnl{eS_n-ex} P^x\lef[ z_n \exp \lef( \e S_n-\e x \ri)\ri] \leq \exp \lef( (\cosh (\e )-1)vn \ri). \edn Since $e^{|t|} \leq e^t +e^{-t}$ ($t \in \R$) and $\cosh (t)-1 \leq \half \cosh (1)t^2$ ($|t| \geq 1$), the desired inequality (\ref{r-expint}) follows from (\ref{eS_n-ex}) by plugging $\pm \e/\sqrt{n}$ in place of $\e$. To prove (\ref{eS_n-ex}), we set $\Phi_n=\exp \lef( \e S_n-\e x \ri)$ and $a_n=P^x[z_n \Phi_n]$. Then, it is easy to see that \bdnn P^x\lef[ z_{k-1}\D \Phi_k \ri] & = & (\cosh (\e )-1)va_{k-1}, \\ P^x[z_{k-1}^0 (e^\b1\{ \D S_k \geq 0 \}-1)\D \Phi_k ] & = & e^{-\e x}\frac{v}{2}(e^\e -1)(e^\b -1)Z^{x,0}_{k-1}, \\ P^x[z_{k-1}^0 \Phi_{k-1} ] & = & e^{-\e x}Z^{x,0}_{k-1}. \ednn Hence we see from (\ref{PFz}) that $$ a_n \leq 1+(\cosh (\e )-1)v\sum_{k=1}^na_{k-1}, $$ which implies (\ref{eS_n-ex}). %%%%%%%%%%%% $\Box$ \vs %%%%%%%%%%%% Proof of \Lem{A_{n,l}}: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We introduce the following notation; \bdnl{M_{m,n}} M_{k,n}=\max_{ k \leq j \leq n}|S_j-S_k|. \edn We divide the proof into two steps; %%%%%%%%%%%%%%%%%%%%%%%%%%% Step1: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We first prove that there is a constant $C_{\ref{M_n/}}=C_{\ref{M_n/}}(\b,v) \in (0,\8)$ such that for all $y>0$ and $\lm \geq 1$; \bdmn \sup_{n \in \N} P^{y\sqrt{vn}}\lef[ z_n : \frac{M_{0,n}}{\sqrt{vn}} \geq \lm \ri] & \leq & C_{\ref{M_n/}}\exp \lef( -\lm /C_{\ref{M_n/}} \ri),\label{M_n/} \\ \sup_{n \in \N}\sqrt{n} P^{0}\lef[ z_n : \frac{M_{0,n}}{\sqrt{vn}} \geq \lm \ri] & \leq & C_{\ref{M_n/}}\exp \lef( -\lm /C_{\ref{M_n/}} \ri).\label{M_n/0} \edmn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We follow Belkin \cite{Bel72}, cf. proof of (2.9) in that paper. We set $A_n=\{ \frac{M_n}{\sqrt{vn}} \geq \lm \}$ and $\t=\inf \lef\{ j \geq 0 \; ; \; \frac{|S_j-S_0|}{\sqrt{vn}} \geq \lm \ri\}$. Then, we have for any $m \geq 1$ that \bdmn \lefteqn{P^{y\sqrt{vn}}\lef[ z_n : A_n \ri] } \nn \\ & = & P^{y\sqrt{vn}}\lef[ z_n : \t \leq n-m \ri] +P^{y\sqrt{vn}}\lef[ z_n : A_n \cap \{ \t > n-m \} \ri]. \label{divA_n} \edmn We divide the first term of (\ref{divA_n}) into \bdmn I_1 & = & P^{y\sqrt{vn}}\lef[ z_n : \t \leq n-m, \frac{S_\t}{\sqrt{vn}} < y-\lm \ri], \nn \\ I_2 & = & P^{y\sqrt{vn}}\lef[ z_n : \t \leq n-m, \frac{S_\t}{\sqrt{vn}} \geq y+\lm \ri]. \nn \edmn Note that $I_1 \neq 0$ only if $y >\lm$, in which case we have $z_\t =1$. Hence \bdmn I_1 & = & P^{y\sqrt{vn}}\lef[ Z^{S_\t}_{n-\t} : \t \leq n-m, \frac{S_\t}{\sqrt{vn}} < y-\lm \ri] \nn \\ & \leq & P^{y\sqrt{vn}}\lef\{ \t \leq n-m, \frac{S_\t}{\sqrt{vn}} < y-\lm \ri\} \nn \\ & \leq & P^{y\sqrt{vn}}\lef\{ \frac{M_n}{\sqrt{vn}} >\lm \ri\} \nn \\ & \leq & C_{\ref{estI1}}\exp (-\lm^2/2). \label{estI1} \edmn To give an upper bound of $I_2$, we observe that there is $C_{\ref{onlyv}} =C_{\ref{onlyv}}(v) \in (1,\8)$ such that \bdnl{onlyv} P^{x\sqrt{vn}}\lef[ z_{n-k}: \frac{S_{n-k}}{\sqrt{vn}} \geq x \ri] \geq 1/C_{\ref{onlyv}} \; \; \mbox{if $x \geq 1$ and $n-k \geq C_{\ref{onlyv}}$}. \edn In fact, \bdmn P^{x\sqrt{vn}}\lef[ z_{n-k}: \frac{S_{n-k}}{\sqrt{vn}} \geq x \ri] & \geq & P^0\lef[ \frac{S^{\min}_{n-k}}{\sqrt{nv}} >-x, \; \frac{S_{n-k}}{\sqrt{vn}} \geq 0 \ri] \nn \\ & \geq & P^0\lef[ \frac{S^{\min}_{n-k}}{\sqrt{v(n-k)}} > -1, \; \frac{S_{n-k}}{\sqrt{v(n-k)}} \geq 0 \ri] \nn \edmn The last displayed probability converges, as $n-k \nearrow \8$, to $$ P\lef\{ \min_{0 \leq t \leq 1}B_t \geq -1, \; B_1 \geq 0 \ri\}>0. $$ This proves (\ref{onlyv}). We now take $m =\lfl C_{\ref{onlyv}} \rfl +1$. We then have by (\ref{p=0}) and (\ref{onlyv}) that $$ Z^{x\sqrt{vn}}_{n-k} \leq 1 \leq C_{\ref{onlyv}}\lef[ z_{n-k}: \frac{S_{n-k}}{\sqrt{vn}} \geq x \ri] \; \; \; \mbox{for $x\geq 1$,} $$ and therefore that \bdmn I_2 & = & \sum_{k \leq n-m}\int_{x \geq y+\lm } P^{y\sqrt{vn}}\lef[ z_k : \t = k, \frac{S_k}{\sqrt{vn}} \in dx \ri] Z^{x\sqrt{vn}}_{n-k} \nn \\ & \leq & C_{\ref{onlyv}} \sum_{k \leq n-m} \int_{x \geq y+\lm } P^{y\sqrt{vn}}\lef[ z_k : \t = k, \frac{S_k}{\sqrt{vn}} \in dx \ri] P^{x\sqrt{vn}}\lef[ z_{n-k}: \frac{S_{n-k}}{\sqrt{vn}} \geq x \ri] \nn \\ & = & C_{\ref{onlyv}} P^{y\sqrt{vn}}\lef[ z_n : \frac{S_n}{\sqrt{vn}} \geq y+\lm \ri]. \label{estI2} \edmn We divide the second term of (\ref{divA_n}) into \bdmn J_1 & = & P^{y\sqrt{vn}}\lef[ z_n : A_n \cap \lef\{ \t > n-m, \frac{|S_n-S_0|}{\sqrt{vn}} > \frac{\lm}{2} \ri\} \ri], \nn \\ J_2 & = & P^{y\sqrt{vn}}\lef[ z_n : A_n \cap \lef\{ \t > n-m, \frac{|S_n-S_0|}{\sqrt{vn}} \leq \frac{\lm}{2} \ri\} \ri]. \nn \edmn We have \bdmn J_2 & = & \int_{x :|x-y| \leq \lm } P^{y\sqrt{vn}}\lef[ z_{n-m} : \frac{S_{n-m}}{\sqrt{vn}} \in dx \ri] P^{x\sqrt{vn}}\lef[ z_{m} : \max_{j \leq n}\lef| \frac{S_j}{\sqrt{vn}}-y\ri| \geq \lm , \; \lef| \frac{S_m}{\sqrt{vn}}-y\ri| \leq \frac{\lm}{2} \ri] \nn \\ & \leq & e^{\b m} \int_{x :|x-y| \leq \lm } P^{y\sqrt{vn}}\lef[ z_{n-m} : \frac{S_{n-m}}{\sqrt{vn}} \in dx \ri] P^{0}\lef[ \frac{M_m}{\sqrt{vn}} \geq \lm\ri] \nn \\ & \leq & C_{\ref{ubJ2}}\exp \lef( -\lm^2n/C_{\ref{ubJ2}}\ri), \label{ubJ2} \edmn where $C_{\ref{ubJ2}}=C_{\ref{ubJ2}}(\b )$. Combining (\ref{estI1}), (\ref{estI2}) and (\ref{ubJ2}), we get \bdnl{estI+J} P^{y\sqrt{vn}}\lef[ z_n : A_n \ri] \leq (1+C_{\ref{onlyv}}) P^{y\sqrt{vn}}\lef[ z_n : \frac{|S_n-S_0|}{\sqrt{vn}} \geq \frac{\lm}{2} \ri] +C_{\ref{ubJ2}}\exp \lef( -\lm^2n/C_{\ref{ubJ2}}\ri). \edn This and (\ref{r-expint}) imply (\ref{M_n/}). On the other hand, we have by (\ref{subf}) that $$ \limn \m^0_n \lef\{ \frac{S_n}{\sqrt{vn}} \geq \frac{\lm}{2} \ri\} =\exp (-\lm^2/8). $$ This, together with (\ref{Znsim}) and (\ref{estI+J}), implies (\ref{M_n/0}). %%%%%%%%%%%% \vs Step2: Proof of (\ref{Mk+l,k}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% It is easy to see that the probability on the left-hand-side of (\ref{Mk+l,k}) is bounded from above by \bdnl{ubMk+l} \sum_{p=0}^{\lfl n/l \rfl}\m_n^0 (A_p), \edn where $A_p=\lef\{ M_{pl, (p+1)l}\geq \lm \sqrt{vl}/3 \ri\}$, cf. \cite[proof of (4.19)]{KaSh91}. For $p \geq 1$, we have by (\ref{p=0}), (\ref{M_n/}) and (\ref{Znsim}) that \bdmn P^0\lef[ z_n : A_p \ri] & \leq & P^0\lef[ z_{(p+1)l} : A_p \ri] \nn \\ & = & \int P^0\lef[ z_{pl} : \frac{S_{pl}}{\sqrt{vpl}}\in dx \ri] P^{x\sqrt{vpl}}\lef[ z_{l} : M_{l}\geq \lm \sqrt{vl}/3\ri] \nn \\ & \leq & C_{\ref{338}}Z^0_{(p+1)l}\exp \lef( -\lm /C_{\ref{338}} \ri) \label{338} \\ & \leq & C_{\ref{ubA_p}}l^{-1/2}\exp \lef( -\lm /C_{\ref{338}} \ri) \label{ubA_p}\edmn For $p=0$, we use (\ref{M_n/0}) to see that \bdnl{339} P^0\lef[ z_n : A_0 \ri] \leq P^0\lef[ z_l : A_0 \ri] \leq C_{\ref{339}}l^{-1/2}\exp \lef( -\lm /C_{\ref{339}} \ri). \edn Therefore, \bdmn \sum_{p=0}^{\lfl n/l \rfl}\m_n^0 (A_p) & \leq & C_{\ref{ubub}} ( n/l )n^{1/2}l^{-1/2}\exp \lef( -\lm /C_{\ref{ubA_p}} \ri) \nn \\ & = & C_{\ref{ubub}}( n/l )^{-3/2}\exp \lef( -\lm /C_{\ref{ubA_p}} \ri). \label{ubub} \edmn This proves (\ref{Mk+l,k}). %%%%%%%%%%%%%%% $\Box$ \vs %%%%%%%%%%%% %%%%%%%%%%%%%%%%%% \SSC{Supercritical phase} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% \subsection{Proof of \Thm{subsup} (b)} %%%%%%%%%%%%%%%%%%%% The augument we persent here is similar to the one developed by van Leeuvan and Hilhorst \cite[Section2]{vLHi81}. Namely, we look at the matrix $Q$ and reduce the original problem to the spectral properties of the matrix. %%%%%%%%%%%%%%%%%%%%% \Lemma{Q-lm} %%%%%%%%%%%%%%%%%%%%% Let $s=1/\lm \in (-1,+1)$. Then, for $\b >\b_c$, an equation \bdnl{Q=lm} Qu=\lm u, \edn given the value of $u(0)$, can uniquely be solved as \bdnl{u(x)} u(x)=\frac{u(0)}{e^\b\sqrt{\gm (s)}} \lef( \tht_{+}(s)\widehat{f^x}(s) +\tht_{-}(s)\widehat{f^x}(s)^{-1}\rig), \; \; \; x \geq 1. \edn In particular, the equation (\ref{Q=lm}) for $|\lm| >1$ has a bounded solution only if $\lm=\lm_{\pm}$. %%%%%%%%%%%%%%% \end{lemma} %%%%%%%%%%%%%%% Proof: The equation (\ref{Q=lm}) reads \bdmn & & e^\b ru(0)+\half e^\b v u(1)=\lm u(0), \label{u(0)} \\ & & ru(x)+\half v\lef( u(x-1)+u(x+1) \rig)=\lm u(x) \; \; \; x \geq 1.\label{u(1)} \edmn If $s=1/\lm \in (-1,+1)$, then the characteristic equation $$ rt+\half v\lef( 1+t^2 \rig)=\lm t $$ for (\ref{u(1)}) has $\widehat{f^{\pm 1}}(s)$ as its two distinct real solutions ($|\widehat{f^1}(s)|<1<|\widehat{f^{-1}}(s)|$). It is therefore elementary to get (\ref{u(x)}). %%%%%%% $\Box$ \vs %%%%%%%%%%%% For $\b >\b_c$, we set \bdnl{rh_b} \rh_2 (x)=\rh (x) \lan \rh, \rh \ran_\b^{-1/2} \; \; \; x\in \N, \edn so that $\lan \rh_2, \rh_2 \ran_\b =1$. %%%%%%%%%%%%%%%%%%%% \Lemma{gap} %%%%%%%%%%%%%%%%%%%%% For $\b >\b_c$, there is a constant $\e_{\ref{gap}}=\e_{\ref{gap}}(\b, v) \in (0,1)$ such that \bdnl{gap} | \lm_{+}^{-n}P^x[f(S_n)z_n] -\lan f,\rh_2 \ran_\b \rh_2 (x)| \leq (1+|x|)^pB_{\ref{gap}}(f) \e_{\ref{gap}}^n \edn for all $x,n \in \N$ and a $p$-bounded function $f: \N \ra \R$, where $B_{\ref{gap}}(f)$ is a constant which depends only on $\b$, $v$ and $f$. %%%%%%%%%%%%%%% \end{lemma} %%%%%%%%%%%%%%% Proof: Since $Q_{xy}=P^x[z_1^y]$, we have \bdmn P^x[f(S_n)z_n] & = & \sum_{y \in \N}P^x[z_n^y]f(y) \nn \\ & = & \sum_{x_1, x_2, \ldots, x_{n-1}, y \in \N} Q_{xx_1}Q_{x_1x_2} \cdots Q_{x_{n-1}y}f(y) \nn \\ & = & Q^nf (x). \label{matrep} \edmn We will prove (\ref{gap}) by studying the matrix $Q$. To do so, let us introduce a Hilbert space $$ \ell^2_\b = \{ f: \N \ra \R \; ; \; \lan f,f \ran_\b <\8 \}, $$ equipped with the inner product defined by (\ref{<>_b}). Let us first prove that \bdnl{gap_2} \| \lm_{+}^{-n}Q^n f-\lan f,\rh_2 \ran_\b \rh_2\|_\b \leq \| f-\lan f,\rh_2 \ran_\b \rh_2\|_\b\e_{\ref{gap_2}}^{n} \; \; \; \mbox{for $f \in \ell^2_\b$,} \edn where $\| \; \cdot \; \|_\b$ denots the $\ell^2_\b$-norm. Consider $Q$ as a bounded operator on $\ell^2_\b$. Then, $Q$ is symmetric and bounded. Therefore, $\s (Q)$: the set of $\ell^2_\b$-spectra of $Q$, is a compact set in $\R$. We claim that \bdmn \s (Q) & \sub & [-1, 1]\cup \{ \lm_{-}, \lm_{+}\}, \label{spec} \\ \lm_{+} & > & \sup\{ |\lm| \in \s (Q) \bsh \{ \lm_{+}\}\}. \label{spegap} \edmn We let $\s^{\rm disc.}(Q)$ and $\s^{\rm ess.}(Q)$ denote respectively, the discrete and the essential spectra of $Q$. Now, consider an operator $$ P=\left( \begin{array}{ccccc} r & \frac{v}2 & 0 & 0 & \cdots\\ \frac{v}2 & r & \frac{v}2 & 0 & \cdots\\ 0 & \frac{v}2 & r & \frac{v}2 & \ddots\\ 0 & 0 & \frac{v}2 & r & \ddots \\ \vdots& \vdots& \ddots& \ddots& \ddots \end{array} \right). $$ Then, $P$ is $\ell^2_0$-symmetric and has the operator norm one on $\ell^2_0$. This implies $\s (P) \sub [-1,1]$, since the set of $\ell^2_\b$-spectra of a bounded operator is independent of the choice of $\b$. Note also that $Q-P$ is of finite rank and hence is a compact operator, which implies that $\s^{\rm ess.}(Q)=\s^{\rm ess.}(P) \sub [-1,1]$, cf. \cite[page 111, Lemma 3]{ReSi78}. On the other hand, we see from \Lem{Q-lm} that $\s^{\rm disc.}(Q) \bsh [-1,+1] \sub \{ \lm_{-}, \lm_{+}\}$. This, together with (\ref{0<1<}), implies (\ref{spec}) and (\ref{spegap}). The claim (\ref{gap_2}) can now easily be seen by (\ref{spec}), (\ref{spegap}) and and the spectral decomposition. The desired estimate (\ref{gap}) for polynomially bounded $f$ can be seen by applying (\ref{gap_2}) to a cut-off function $f_{x,n} (y)=f(y)1\{ |x-y| \leq n\}$ as follows. Since $Q^nf_{x,n} (x)=Q^nf (x)$, we have \bdmn \lefteqn{|\lm_{+}^{-n}Q^n f(x)-\lan f,\rh_2 \ran_\b \rh_2 (x)|} \nn \\ & \leq & |\lm_{+}^{-n}Q^n f_{x,n}(x)-\lan f_{x,n},\rh_2 \ran_\b \rh_2 (x)| +|\lan f_{x,n}-f,\rh_2 \ran_\b |\rh_2 (x) \label{f_{x,n}} \edmn Since $f_{x,n} \in \ell^2_\b$, we see from (\ref{gap_2}) that the first term of (\ref{f_{x,n}}) is not greater than $ \e_{\ref{gap_2}}^{n} \| f_{x,n} \|_\b$ which has a bound in the form of the right-hand-side of (\ref{gap}). On the other hand, it is easy to bound the second term of (\ref{f_{x,n}}) by $B(f)\rh (n)$, where $B(f)$ is a constant which depends only on $\b$, $v$ and $f$. %%%%%%%% $\Box$ \vs %%%%%%%%% We prove (\ref{sup1}) in the following generalized form. %%%%%%%%%%%%%%%%%%%% \Lemma{Fh} %%%%%%%%%%%%%%%%%%%%% For $\b >\b_c$, there are constants $C_{\ref{Fh}}=C_{\ref{Fh}}(\b, v) >0$ and $\e_{\ref{Fh}}=\e_{\ref{Fh}}(\b, v) \in (0,1)$ as follows; if $l \geq k +(1+x)C_{\ref{Fh}}$, $f: \N^k \ra \R $ and $h: \N \ra \R$ are polynomially bounded, $F=f(S_1, \ldots, S_k)$, then \bdnl{Fh} \lef| \m^x_l[Fh(S_l)] -\frac{\lan h, \rh_2\ran_\b }{\lan 1, \rh_2\ran_\b }\m^x_l[F] \rig| \leq B_{\ref{Fh}}(f,h)\e_{\ref{Fh}}^{l-k}. \edn where $B_{\ref{Fh}}(f,h)$ is a constant which depends only on $\b$, $v$, $f$ and $g$. %%%%%%%%%%%% \end{lemma} %%%%%%%%%%%%%% Proof : %%%%%%%%%%% Since $f$ is polynomially bounded, it is easy to see that there are constants $p>0$ and $B_{\ref{P^xpb}}=B_{\ref{P^xpb}}(\b, v, f)>0$ such that \bdn \sup_{k,x \in \N }(1+|x|)^{-p} P^x[|F |z_k]\lm_{+}^{-k} \leq B_{\ref{P^xpb}}, \label{P^xpb} \\ \edn It is also easy to see from (\ref{gap}) that there is a constant $C_{\ref{Zgeqrh}}=C_{\ref{Zgeqrh}}(\b, v)>0$ such that \bdnl{Zgeqrh} 1/2 \leq \frac{\lm_{+}^{-n}Z^x_n}{\lan 1, \rh_2 \ran_\b \rh_2 (x)} \leq 3/2 \; \; \; \mbox{if $n \geq (1+x)C_{\ref{Zgeqrh}}$.} \edn We take $C_{\ref{Fh}} \geq C_{\ref{Zgeqrh}}$. We see from (\ref{gap}) that \bdmn \lefteqn{\lef| P^y[h(S_l)z_l] -\frac{\lan h, \rh_2\ran_\b }{\lan 1, \rh_2\ran_\b }Z^y_l \rig| } \nn \\ & \leq & \lm_{+}^{l}\lef| \lm_{+}^{-l}P^y[h(S_l)z_l]-\lan h, \rh_2\ran_\b \rh_2 (y) \rig| +\lm_{+}^{l}\lef| \frac{\lan h, \rh_2\ran_\b }{\lan 1, \rh_2\ran_\b } \lef( \lm_{+}^{-l}Z^y_l-\lan 1, \rh_2\ran_\b \rh_2 (y) \rig) \rig| \nn \\ & \leq & (1+|y|)^pB_{\ref{agap}}\lm_{+}^{l}\e_{\ref{gap}}^l, \label{agap} \edmn for some $p=p(h) \geq 0$ and $B_{\ref{agap}}=B_{\ref{agap}}(\b, v, g)>0$. We have on the other hand that \bdmn \lefteqn{\m_l^x[Fh(S_l)] } \nn \\ & = & \frac{1}{Z^x_l}\sum_{y \in \N}P^x[Fz^y_k]P^y[h(S_{l-k})z_{l-k}] \nn \\ & = & \frac{1}{Z^x_l}\sum_{y \in \N}P^x[Fz^y_k] \lef( P^y[h(S_{l-k})z_{l-k}] -\frac{\lan h, \rh_2\ran_\b }{\lan 1, \rh_2\ran_\b }Z^y_{l-k}\rig) +\frac{\lan h, \rh_2\ran_\b }{\lan 1, \rh_2\ran_\b }\m_l^x[F]. \label{m[Fh]} \edmn By (\ref{agap}) and (\ref{m[Fh]}), \bdmn \lefteqn{\lef|\m^x_l[Fh(S_l)] -\frac{\lan h, \rh_2\ran_\b }{\lan 1, \rh_2\ran_\b }\m^x_l[F] \rig|} \nn \\ & \leq & B_{\ref{agap}} \frac{\lm_{+}^{-k}P^x_k[|F|(1+|S_k|)^pz_k]}{\lm_{+}^{-l}Z^x_l} \e_{\ref{gap}}^{l-k} \nn \edmn By (\ref{Zgeqrh}) and (\ref{P^xpb}), this is bounded from above by \bdnl{agapa} B_{\ref{agapa}}\e_{\ref{agapa}}^{l-k} \edn for some $B_{\ref{agapa}}=B_{\ref{agapa}}(\b, v, f, g)>0$ and $\e_{\ref{agapa}}=\e_{\ref{agapa}}(\b, v)\in (0,1)$ if $C_{\ref{Fh}}$ is large enough. This completes the proof of the lemma. %%%%%%%% $\Box$ %%%%%%%%% \subsection{Proof of \Thm{sup}} %%%%%%%%%%%% Since $f$ is polynomially bounded, it is easy to see that there are constants $p>0$ and $B_{\ref{P^xpb2}}=B_{\ref{P^xpb2}}(\b, v, f)>0$ such that \bdn \sup_{k,x \in \N }(1+|x|)^{-p} \m_k^x[|F |] \leq B_{\ref{P^xpb2}}. \label{P^xpb2} \edn We next observe that there are constants $C_{\ref{obs}}=C_{\ref{obs}}(\b,v)>0$ and $\e_{\ref{obs}}=\e_{\ref{obs}}(\b,v)\in (0,1)$ such that if $n \geq k+(1+x)C_{\ref{obs}}$, then \bdnl{obs} \lef|\frac{\lm_+^{-(n-k)}Z^y_{n-k}}{\lm_{+}^{-n}Z^x_n} -\frac{\rh (y)}{\rh(x)} \rig| \leq C_{\ref{obs}}\e_{\ref{obs}}^{n-k} \; \; \; \mbox{for all $y \in \N$.} \edn This can be seen by (\ref{gap}). We have \bdmn \m_n^x[F] & = & (Z^x_n)^{-1}\sum_{y \in \N}P^x[Fz^y_k]Z^y_{n-k} \nn \\ & = & \sum_{y \in \N}P^x[Fz^y_k]\lm_{+}^{-k} \lef( \frac{\lm_+^{-(n-k)}Z^y_{n-k}}{\lm_{+}^{-n}Z^x_n} -\frac{\rh (y)}{\rh (x)}\rig) +\lm_{+}^{-k}\sum_{y \in \N}P^x[Fz^y_k]\frac{\rh (y)}{\rh (x)}. \label{m_n^xF} \edmn Since the second term of (\ref{m_n^xF}) equals $\n^x[F]$, we see from (\ref{obs}), (\ref{m_n^xF}) and (\ref{P^xpb}) that \bdnn |\m_n^x[F] - \n^x[F]| & \leq & C_{\ref{obs}}\lm_{+}^{-k}|P^x[Fz_k]| \e_{\ref{obs}}^{n-k} \nn \\ & \leq & C_{\ref{obs}}B_{\ref{P^xpb}}(1+|x|)^p \e_{\ref{obs}}^{n-k} \ednn We therefore get (\ref{muranu}) by taking $C_{\ref{muranu}}$ large enough. %%%%%%%%% $\Box$ %%%%%%%%%%%% %%%%%%%%%%%% \subsection{ Proof of \Thm{decay}} %%%%%%%%%%%%% It is clear that (\ref{nudec}) follows from (\ref{muranu}) and (\ref{decay}). It is therefore, enough to prove (\ref{decay}). We set $h(y)=h_{n,l}(y)=P^y[z_{n-l}g(S_1, \ldots, S_m)]\lm_{+}^{-(n-l)}$. Then, by (\ref{P^xpb}), $h$ is polynomially bounded, uniformly in $n \geq 1$ and $1 \leq l \leq n $. It is easy to see that $$ \m^x_n[FG] = \frac{\lm_{+}^{-l}Z^x_l}{\lm_{+}^{-n}Z^x_n}\m^x_l[Fh(S_l)] \; \; \mbox{and} \; \; \m^x_n[G] = \frac{\lm_{+}^{-l}Z^x_l}{\lm_{+}^{-n}Z^x_n}\m^x_l[h(S_l)]. $$ Therefore, \bdmn \lefteqn{\mu_n^x (F\; G) -\mu_n^x (F)\mu_n^x (G)} \nn \\ & = & \frac{\lm_{+}^{-l}Z^x_l}{\lm_{+}^{-n}Z^x_n} \lef( \m^x_l[Fh(S_l)]-\m^x_l[F]\m^x_l[h(S_l)]\rig) \label{;dec1} \\ & + & \frac{\lm_{+}^{-l}Z^x_l}{\lm_{+}^{-n}Z^x_n} \m^x_l[h(S_l)]\lef( \m^x_l[F]-\m^x_n[F]\rig). \label{;dec2} \edmn We see from (\ref{muranu}), (\ref{Zgeqrh}) and (\ref{P^xpb}) that if $C_{\ref{decay}}$ is large enough, then the absolute value of (\ref{;dec2}) is not greater than \bdnl{;deco2} B_{\ref{;deco2}}\e_{\ref{muranu}}^{l-k}, \edn for some $B_{\ref{;deco2}}=B_{\ref{;deco2}}(\b, v, f, g)>0$. We now turn to (\ref{;dec1}). We write \bdmn \lefteqn{\m^x_l[Fh(S_l)]-\m^x_l[F]\m^x_l[h(S_l)]} \nn \\ & = & \lef( \m^x_l[Fh(S_l)] -\frac{\lan h, \rh_2\ran_\b }{\lan 1, \rh_2\ran_\b }\m^x_l[F] \rig) +\m^x_l[F] \lef( \frac{\lan h, \rh_2\ran_\b }{\lan 1, \rh_2\ran_\b } -\m^x_l[h(S_l)]\rig).\label{wrote} \edmn We see from (\ref{Fh}) and (\ref{P^xpb2}) that the both terms of (\ref{wrote}) has the absolute value not greater than \bdnl{aagap} B_{\ref{aagap}}\e_{\ref{sup1}}^l \edn for some $B_{\ref{aagap}}=B_{\ref{aagap}}(\b, v, f,g)>0$ if $C_{\ref{decay}}$ is large enough. %%%%%%%% $\Box$ %%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%% \footnotesize %%%%%%%%%%%%%%%% \vvs {\it Acknowledgement:} The authors thank Professor Y. Takahashi for suggesting the compact perturbation augument used in \Lem{gap}. We also thank Professor S. 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