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random walk, pinning, wetting, limit theorem
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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 <i_m =n}
P^x[h_n: T_1=i_1, \ldots, T_m=i_m] \nn \\
& = & 
\sum_{n \geq 1}s^n\sum_{m \geq 1}e^{(m-1)\b}
\sum_{\barray{l} 
\mbox{\scriptsize $j_1 \geq 1, \ldots, j_m \geq 1$} \\
\mbox{\scriptsize $j_1+\ldots +j_m=n$} \earray}
g^x(j_1)g^0(j_2) \cdots g^0(j_m)\nn \\
& = & 
\sum_{m \geq 1}\sum_{j_1 \geq 1} \ldots \sum_{j_m \geq 1}
s^{j_1}g^x(j_1)\lef( e^{\b}s^{j_2} g^0(j_2) \rig) 
\cdots \lef( e^{\b} s^{j_m}g^0(j_m) \rig) \nn \\
& = &  
\widehat{g^x}(s)\sum_{m \geq 1}
\lef( e^{\b} \widehat{g^0}(s) \rig)^{m-1}\nn \\
& = &  
\widehat{g^x}(s)\sum_{m \geq 1}
\lef( 1-\tht_{-}(s)\rig)^{m-1},
\label{P^x0}
\edmn 
which, implies (\ref{^Z^x0}) for the case $x \geq 1$. 
Equality (\ref{^Z^x0}) for the case $x=0$ can be seen 
 similarly as above just by replacing 
$e^{(m-1)\b}$ and $g^x(j_1)$ respectively by 
$e^{m\b}$ and $g^0(j_1)$.
%%%%%%%%%%%%
$\Box$
%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
\Lemma{^Z^x}
%%%%%%%%%%%%%%%%%%%%%
For $0 \leq s <1$, 
\bdnl{^Z^x}
\widehat{Z}^{x}(s)
=\lef( 1- \del^{x,0} \rig)\frac{1-\widehat{f^x}(s)}{1-s}
+\lef( \half e^\b\frac{1-\widehat{f^0}(s)}{1-s}-\half e^\b+1 \rig) 
\widehat{Z}^{x,0}(s).
\edn 
In particular, 
\bdnl{^Z^x1}
\widehat{Z}^{x}(s)
\st{s \nearrow 1}{\sim}
C_{\ref{^Z^x1}}/(1-s)^{1/2}
\; \; \; \mbox{if $\b <\b_c$,}
\edn 
where 
$C_{\ref{^Z^x1}}=\sqrt{\frac{2}{v}}
\lef( x+\frac{ve^\b}{2\tht_{-} (1)}\rig)$.
%%%%%%%%%%%%%%%
\end{lemma}
%%%%%%%%%%%%%%%
Proof: 
We begin by showing that 
\bdnl{ZZ}
Z^x_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, 
\edn
where $F^x_n=P^x\{ T_1 >n \}$. 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 $k<n$,
\bdmn 
\mbox{If } S_{k-1}>0,&& \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<t,s<1;|s-t|<\delta} |W(vns)-W(vnt)| >\e \sqrt{n}]<\e .
\edn
Now it remains to put these together:
\bdnn
\sup_{0<t<1} \left|\frac{W(vnt)}{\sqrt{n}}-\frac{\tilde S_{nt}}{\sqrt{n}}\right| 
&<&\e + \sup_{0\leq k\leq n} \left|\frac{W(vk)}{\sqrt{n}}-\frac{S_k}{\sqrt{n}}\right| \\
&\equiv& \e + \sup_{0\leq k\leq n} \left|\frac{W(vk)}{\sqrt{n}}-\frac{W(\s_{k})}{\sqrt{n}}\right| \\
&<& 2\e + \sup_{0\leq k\leq n} \left|\frac{W(vk)}{\sqrt{n}}-\frac{W(\sum_{ j\leq k; W(\s_{j-1})\neq0} (\s_{j}-\s_{j-1}))}{\sqrt{n}}\right| \\
&<& 3\e + \sup_{0\leq k\leq n} \left|\frac{W(vk)}{\sqrt{n}}-\frac{W(\sum_{ j\leq k; W(\s_{j-1})\neq0} v)}{\sqrt{n}}\right| \\
&<&4\e
\ednn
on the intersection of the events that appeared in
(\ref{ByKolmIneq}),
(\ref{Visit-0-Mean}),
(\ref{Visit-0-RV}) and (\ref{Conti-W}).

We have thus proven $\tl{\m}_n$ converges weakly 
to the law of
$\{ W(vnt)/\sqrt{n}; 0\leq t\leq 1\}$
which in turn is identical to that of
$\{ \sqrt{v} W(t); 0\leq t\leq 1\}$.
%%%%%%%%%%%%%
$\Box $
%%%%%%%%%%%%%
\subsection{Convergence of finite dimensional distributions for $\b <\b_c$}
%%%%%%%%%%%
The purpose of this subsection is to prove the following 
%%%%%%%%%%%%%%%%%%%%
\Lemma{fdd}
%%%%%%%%%%%%%%%%%%%%%
Let $\b<\b_c$, $k \geq 1$ and 
$0<t_1 < \ldots, <t_k \leq 1$. 
Suppose that $\{ t_{j,n} >0\; ; \; 1 \leq j \leq k, \; n \geq 1\}$ 
are such that $0<t_{1,n} < \ldots, <t_{k,n} $ for all $n \geq 1$ and 
that $\sup_{n \geq 1}|t_{j,n}-nt_j|<\8 $ for all $1 \leq j \leq k$. 
Then, probability measures on $\R^k$ defined by 
\bdnl{fddvect}
\m_n^0\lef\{ \lef(S_{t_{j,n}}/\sqrt{vn} \rig)^k_{j=1} \in \cdot \rig\},
\; \; \ ; n=1,2,\ldots 
\edn 
converge weakly to the law of $\lef(W_{t_j} \rig)^k_{j=1}$.
%%%%%%%%%%%%%%%
\end{lemma}
%%%%%%%%%%%%%%
We introduce the following functions;
\bdmn
g_t (x) & = & (2\pi t)^{-1}\exp \lef( -\frac{x^2}{2t}\rig),
\; \; \; t>0, \; 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 $0 \leq s <t \leq 1$ and $x,y \geq 0$
\bdnl{p(s,x|t,y)}
p(s,x|t,y)=
\lef\{ \barray{ll}
t^{-3/2}y\exp \lef(-\frac{y^2}{2t}\rig)h(t,y)
& \mbox{if $s=0$ and $x=0$}\\
h(s,x)^{-1}g_{t-s} (x,y) h(t,y)
& \mbox{if $s>0$ 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 $0 \leq s <t <1$, $x, y \geq 0$, 
\bdmn
\limn P^{y\sqrt{vn}}\lef[ z_{n-\lfl ns \rfl}: 
\frac{S_{\lfl nt \rfl-\lfl ns \rfl}}{\sqrt{vn}} \leq x\ri]
& = & P\{ \min_{0 \leq u \leq 1-s}B_u \geq -y, \;  B_{t-s} \geq x-y \} 
\label{limhp}\\
& = & h(s,y)\int^x_0p(s,y|t,z). \nn 
\edmn 
The convergence is uniform in $(x,y) \in [0,\8) \times [0,b]$ 
for any $b>0$. 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. Watanabe for bringing 
a work of Iglehart \cite{Igl74} into our interest. 


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\end{document}
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