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\centerline{\bf ALMOST SURE FUNCTIONAL LIMIT THEOREMS}
\centerline{Part I. The general case}\smallskip
\centerline{\it P\'eter Major}
\centerline{Mathematical Institute of the Hungarian Academy of Sciences}
\centerline {and}
\centerline{Bolyai College of E\"otv\"os Lor\'and University, Budapest}
\medskip{\narrower{\narrower In this paper we formulate and prove
the almost sure functional limit theorem in fairly general cases. This
limit theorem is a result which states that if a stochastic process
$X(t,\oo)$, $t\ge0$, is given on a probability space with some nice
properties, then an appropriate probability measure $\bar\lambda_T$
can be defined on the interval $[1,T]$ for all $T>1$ in such a way
that for almost all $\oo$ the distributions of the appropriate
normalizations of the trajectories $X_t(\cdot,\oo)=X(t\cdot,\oo)$,
considered as random variables $\xi_T(t)$, $t\in[1,T]$, on the
probability spaces $([1,T],\Cal A,\lambda_T)$ with values in a
function space have a weak limit independent of $\oo$ as
$T\to\infty$. We shall consider self-similar processes which appear
in different limit theorems. The almost sure functional limit theorem
will be formulated and proved for them and their appropriate
discretization under weak conditions. We also formulate and prove a
coupling argument which makes it possible to prove the almost sure
functional limit theorem for certain processes which converge to a
self-similar process. In the second part of this work we shall prove
and generalize --- with the help of the results of the first part ---
some known almost sure functional limit theorems for independent
random variables. \par}\par}
\beginsection 1. Introduction
The following ``almost sure central limit theorem" is a popular
subject in recent research. Let $X_1(\oo),X_2(\oo),\dots$ be a sequence
of iid.\ random variables, $EX_1=0$, $EX_1^2=1$,
$S_n(\oo)=\summ_{k=1}^n X_k(\oo)$ on a probability space $(\Omega,\Cal
A,P)$. (In the sequel we denote by $(\Omega,\Cal A,P)$ the probability
space where the random variables we are considering exist.) Then
$$
\lim_{n\to\infty}\frac1{\log
n}\sum_{k=1}^n\frac1kI\left(\frac{S_k(\oo)}{\sqrt k}__0$, if
$$
X(u,\oo)\overset \Delta \to =\frac {X(Tu,\oo)}{T^{1/\alpha}},\quad 0\le
u<\infty, \tag2.1
$$
for all $T>0$, where $\overset\Delta\to=$ means that the processes
at the two sides of the equation have the same distribution. (Here we
consider the distribution of the whole process $X(u,\oo)$, $u\ge0$, and
not only its one-dimensional distributions.)}
\medskip
The Wiener process is self-similar with self-similarity parameter
$\alpha=2$. Similarly, for all stable laws $G$ with
parameter $\alpha$, $0<\alpha<2$, $\alpha\neq1$, a so-called stable
process $X(u,\oo)$ can be constructed which has independent and
stationary increments, $X(0,\oo)\equiv0$, which is self-similar with
self-similarity parameter $\alpha$, and the distribution function
of $X(1,\oo)$ is $G$. The case $\alpha=1$ is exceptional. In this case
(except the special case when $X(1,\oo)$ has symmetric distribution)
only a modified
version of formula (2.1) holds, where a norming factor $\const\log T$
must be added with an appropriate non-zero constant to one side in
formula (2.1). Another example for self-similar processes was given by
Dobrushin in paper [3], who could construct new kind of self-similar
processes subordinated to a Gaussian process. He constructed them by
working with non-linear functionals of Gaussian processes.
Now we introduce the following notion:
\medskip\noindent
{\bf Definition of generalized Ornstein--Uhlenbeck processes.} {\it
Let $X(u,\oo)$, $u\ge0$, be a self-similar process with self-similarity
parameter $\alpha>0$. We call the process $Z(t,\oo)$,
$-\infty0$. Hence the Wiener process
$W(t,\oo)$ and any of its scaled version $A_TW(Tt,\oo)$, $0\le t\le 1$,
where $T>0$ and $A_T>0$ are arbitrary constants, can be considered as
random variables taking values in the space $C([0,1])$ of continuous
functions on the interval $[0,1]$. The processes $X(t,\oo)$,
$A_TX(tT,\oo)$, $0\le t\le1$, where $X(t,\oo)$, $0\le t<\infty$, is
a stable process, can be considered as random variables on the space
$D([0,1])$ of c\`adl\`ag functions on the interval $[0,1]$.
We shall work not only in the space $C([0,1])$ but also in the space
$D([0,1])$. To work in the space $D([0,1])$ one has to handle some
unpleasant technical problems. But since we also want to investigate
stable processes in Part~II.\ of this work, we also have to work
in this space. We shall apply the book of P.~Billingsley~[1] as the
main reference for this subject.
We consider both spaces $C([0,1])$ and $D([0,1])$ with the usual
topology, and the Borel $\sigma$-algebra generated by this topology.
Both spaces can be endowed with a metric which induces this topology,
and with which these spaces are separable, complete metric spaces. A
detailed discussion and proof of these results and definitions can be
found in the book of P.~Billingsley~[1]. Since we shall need the exact
form of these metrics we recall these results. In the $C([0,1])$ space
the supremum metric $\rho(x,y)=\supp_{0\le t\le1}|x(t)-x(s)|$ is
considered. In the space $D([0,1])$ the following metric
$d_0(\cdot,\cdot)$ satisfies these properties: For a pair of functions
$x,y\in D([0,1])$ $d_0(x,y)\le\e$, if there exists such a homeomorphism
$\lambda(t)\:[0,1]\to [0,1]$ of the interval $[0,1]$ into itself for
which $\lambda(0)=0$, $\supp_{t\neq
s}\log \left|\dfrac{\lambda(t)-\lambda(s)}{t-s}\right|\le\e$,
and $|x(t)-y(\lambda(t))|\le\e$ for all $t\in [0,1]$. (See for instance
Theorems 14.1 and~14.2 in Billingsley's book~[1].) In the sequel we
shall apply these metrics in the spaces $C([0,1])$ and $D([0,1])$, and
denote them by $\rho(\cdot,\cdot)$.
Let us also recall that given some probability measures $\mu_T$ on a
metric space $\bold K$ indexed by $T\in [1,\infty)$ or
$T=\{A_1,A_2,\dots\}$, $\limm_{n\to\infty}A_N=\infty$, the measures
$\mu_T$ converge weakly to a measure $\mu$ on $\bold K$ as $T\to\infty$
if $\limm_{T\to\infty}\intl_{\bold K}\Cal F(x)\mu_T(\,dx)= \intl_{\bold
K}\Cal F(x)\mu(\,dx)$ for all continuous and bounded functionals
$\Cal F$ on the space $\bold K$. The next result states the almost sure
functional limit theorem for a self-similar process which satisfies some
additional conditions. The proof is based on the ergodic theorem applied
for the generalized Ornstein--Uhlenbeck process corresponding to this
self-similar process.
\medskip\noindent
{\bf Theorem 1.} {\it Let $X(u,\oo)$ be a self-similar process with
continuous or c\`adl\`ag trajectories, and $Z(t,\oo)$ the generalized
Ornstein--Uhlenbeck process corresponding to it. The process $Z(t,\oo)$,
$-\infty0.
\tag2.4
$$
Let us define for all $\oo\in \Omega$ and $T\ge 1$ the (random)
probability measure $\mu_T(\oo)$ in the space $C([0,1])$ or $D([0,1])$
which is concentrated on the trajectories $X_t(\oo)$, $1\le t\le T$,
and takes the value $X_t(\oo)$, $1\le t\le T$, with probability
$\dfrac1{\log T}\dfrac{dt}t$. More formally, for a measurable set
$\bold A\subset C([0,1])$ or $\bold A\subset D([0,1])$ put
$\mu_T(\oo)(\bold A)=\bar \lambda_T\{t\:X_t(\oo)\in \bold A\}$, where
$\bar\lambda_T$ is a measure on $[1,T]$ defined by the formula
$\lambda_T(\bold C)=\dfrac1{\log T}\intl_{\bold C}\,\dfrac{dt}t$ for
all measurable sets $\bold C\subset[0,T]$.
The following version of Formula {\rm (2.3)} also holds:
For almost all $\oo\in\Omega$ the probability measures
$\mu_T(\oo)$ converge weakly to the distribution of the process
$X_1(u,\oo)$ defined in {\rm (2.4)} with $t=1$, or in other words, there
is a set of probability one such that if $\oo$ is in this set then
relation {\rm (2.3)} holds for this $\oo$ and \/ {\rm all} bounded and
continuous functionals~$\Cal F$.
If $X(u,\oo)$ is a Wiener or stable process, then the generalized
Ornstein--Uhlenbeck process corresponding to it is not only stationary,
but also ergodic. Hence the results of Theorem~1 are applicable in this
case.}
\medskip
We want to prove a discretized version of the above result, where the
measures $\mu_T(\oo)$ concentrated in the set of trajectories
$X_t(\oo)$, $1\le t\le T$, are replaced by some measures $\mu_N(\oo)$
which are concentrated on a set of trajectories $X_{a(j,N)}(\oo)$ with
appropriate weights, and the numbers $a(j,N)$ constitute a finite set.
Then we want to make a further discretization, where the trajectories
$X_{a(j,N)}$ are replaced by their discretized version. To prove these
results in the case when the trajectories of the process $X(\cdot,\oo)$
are c\`adl\`ag functions we impose the following additional condition.
$$
P\(\lim_{t\to 1-0} X(t,\oo)=X(1,\oo)\)=1. \tag2.5
$$
First we formulate a result which serves as the basis of the
discretization results formulated later.
\medskip\noindent
{\bf Theorem 2.} {\it Let $X(u,\oo)$, $X_t(u,\oo)$, $\mu_T(\oo)$ and
$\mu_0$ be the same as in Theorem~1. Let us assume that the conditions
of Theorem~1 are satisfied, and also the additional condition {\rm
(2.5)} holds in the case when the process $X(\cdot,\oo)$ has c\`adl\`ag
trajectories.
Let us define, similarly to the trajectories $X_t(\cdot,\oo)$ defined in
{\rm (2.4)}, the following transformed functions $x_t=x_t(\cdot)$ of a
function $x\in C([0,1])$ or $x\in D([0,1])$ by the formula
$$
x_t(u)=x_{t,\alpha}(u)=t^{-1/\alpha}x(ut), \quad 0\le u\le1,\;0\delta\)=0 \quad\text{for all }\delta>0. \tag2.6
$$
where $\rho(\cdot,\cdot)$ is the metric whose definition was recalled
before Theorem~1, and with which $C(0,1])$ or $D([0,1])$ are separable,
complete metric spaces. (Let us recall that the (random) measure
$\mu_T(\oo)$ is concentrated on the trajectories $X_u(\cdot,\oo)$,
$1\le u\le T$, of the process $X(\cdot,\oo)$ defined by
formula~{\rm(2.4).)}}
\medskip
Condition (2.5) had to be imposed to control the behaviour of the
trajectories of the processes $X_t(u,\oo)$ in the end point $u=1$. This
is not a strict restriction. For instance the next simple Lemma~1 gives
a sufficient condition for its validity. It implies in particular, that
the stable processes with self-similarity parameter $\alpha$,
$0<\alpha<2$, $\alpha\neq1$, satisfy relation (2.5).
\medskip\noindent
{\bf Lemma 1.} {\it Let $X(\cdot,\oo)$ be a self-similar process
with self-similarity parameter $\alpha>0$ which is also a process with
stationary increments, and whose trajectories are c\`adl\`ag functions.
Then it satisfies relation {\rm (2.5).}}
\medskip
Now we formulate the result about ``possible discretization"
of the measures $\mu_T$ in the result of Theorem~1.
Before this we make some comments which can explain the
content of this result.
For all $T>1$ let us consider the probability space $([1,T],\Cal A,\bar
\lambda_T)$, where $\Cal A$ is the Borel $\sigma$-algebra, and
$\bar\lambda_T$ is the measure defined in the formulation of Lemma~1.
Fix an $\oo\in\Omega$, and let us consider the random variable $\xi(t)$,
$1\le t\le T$, as $\xi(t)=X_t(\cdot,\oo)$, defined in formula (2.4), in
the probability space $([1,T],\Cal A,\bar\lambda_T)$. This is a random
variable which takes its value in the space $C([0,1])$ or $D([0,1])$,
and it has distribution $\mu_T(\oo)$. Let us consider the above
construction with some $T=B_N$, together with a dense splitting
$1=B_{N,1}0$ restricted to the interval $[0,1]$, and $A_n$ is chosen as
$A_n=B_n^{1/\alpha}$.
Let us remark that if the random variables $\xi_k(\oo)$ satisfy the
almost sure functional central limit theorem with weight functions
$A_n=\sqrt n$ and $B_n=n$, --- and in Part II. we shall prove that under
the conditions imposed for the validity of formula (1.1) this is the
case, --- then they also satisfy relation (1.1). To see this, fix a
real number $u$ and define the functional $\Cal F=\Cal F_t$ in the
space $C([0,1])$ by the formula $\Cal F(x)=1$ if $x(1)< u$, and $\Cal
F(x)=0$ if $x(1)\ge u$, where $x\in C([0,1])$, i.e.\ it is a continuous
function on the interval $[0,1]$. This functional $\Cal F$ is
continuous with probability one with respect to the Wiener measure
$\mu_0$. Hence $\intl\Cal F(x)\,d\mu_n(\oo)(x)\to
\intl\Cal F(x)\,d\mu_0(x)$ for almost all $\oo$. This relation is
equivalent to formula (1.1). Indeed, the right-hand side of this
relation equals the right-hand side of formula (1.1),
while the left-hand side is a slight modification of the left-hand
side of $(1.1)$. The difference between these formulas is that the
weights $\dfrac1k$ in (1.1) are replaced by $\log\dfrac{k+1}k$ in the
other formula, and summation goes from 1 to $n-1$ instead of summation
from 1 to $n$. Since $\log\dfrac{k+1}k=\dfrac1k+O\(\dfrac1{k^2}\)$
these two relations are equivalent.
We formulate the following statement because of its importance in
later applications in form of a Corollary.
\medskip\noindent
{\bf Corollary.} {\it Let $X(\cdot,\oo)$ be a self-similar process
with self-similarity parameter $\alpha>0$ such that its trajectories
are in the $C([0,1])$ or $D([0,1])$ space, it satisfies relation
(2.5), and the generalized Ornstein--Uhlenbeck process corresponding
to it is ergodic. Let $t_n$, $n=0,1,\dots$, $t_0=0$, be an increasing
sequence of real numbers such that $\limm_{n\to\infty}t_n=\infty$,
$\limm_{n\to\infty}\dfrac{t_{n+1}}{t_n}=1$. Put $\eta_n(\oo)
=X(t_n,\oo)-X(t_{n-1},\oo)$, $B_n=t_n$, $A_n=B_n^{1/\alpha}$,
$n=1,2,\dots$. Then the sequence $\eta_n(\oo)$, $n=1,2,\dots$,
satisfies the almost sure functional limit theorem with
weight functions $A_n$ and $B_n$ and limit measure $\mu_0$ which is
the distribution of the process $X(u,\oo)$, restricted to $0\le u\le
1$.}
\medskip
To prove this Corollary define the process
$X'(u,\oo)=A_1^{-1}X(B_1u,\oo)$ and observe that it has the
same distribution as the process $X(u,\oo)$. Define the real numbers
$B_{k,N}=\dfrac {t_k}{t_1}$, $1\le k\le N$, consider the random
broken lines $\bar X'_{B_{j,N}}(\cdot,\oo)$, $1\le j\le N$, and the
random measure $\bar \mu_N(\oo)$ defined in the formulation of
Theorem~3 with this process $X'(\cdot,\oo)$ and these numbers
$B_{k,N}$, (with the choice $k_N=N$), and apply Theorem~3, --- whose
conditions are satisfied, --- for these random measures
$\bar\mu_N(\oo)$.
On the other hand, define the random broken lines $S_k(s,\oo)$ by
formula (2.11) with $B_N=t_N$, $A_N=B_n^{1/\alpha}$ and the partial sums
$S_k(\oo)=\summ_{l=1}^k(X(t_l,\oo)-X(t_{l-1},\oo))$, and let us also
define the measure $\mu_N(\oo)$ by formula (2.12) with these random
broken lines. Then a comparison shows that the above defined broken
lines $\bar X'_{B_{j,N}}(\cdot,\oo)$ and $S_j(\cdot,\oo)$ and also their
distributions, the random measures $\bar\mu_N(\oo)$ and
$\mu_N(\oo)$ agree. Hence the second statement of Theorem 3 implies the
almost sure functional limit theorem in this case.
If a sequence of random variables $\xi_n(\oo)$, $n=1,2,\dots$, is close
to this sequence $\eta_n(\oo)$, then it is natural to except that this
new sequence satisfies the same almost sure functional limit theorem.
We want to give a good coupling argument that enables us to prove this
for a large class of processes $\xi_n(\oo)$. For this aim we define a
Property~A. We prove that if Property~A holds for a pair of sequences
or random variables $(\xi_n(\oo),\eta_n(\oo))$, $n=1,2,\dots$, and the
sequence $\eta_n(\oo)$, $n=1,2,\dots$, satisfies the almost sure
functional limit theorem, then the sequence $\xi_n(\oo)$, $n=1,2,\dots$
also satisfies the almost sure functional limit theorem with the same
norming constants and limit law.
\medskip\noindent
{\bf Definition of Property A.} {\it Let $\eta_n(\oo)$, $n=1,2,\dots$,
be a sequence
of random variables which satisfies the almost sure functional limit
theorem with a limit measure $\mu_0$ in the space $C([0,1])$ or
$D([0,1])$ and some weight functions $A_n$ and $B_n$ satisfying
relation (2.10). Let us also assume that the limit measure $\mu_0$ is
the distribution of the restriction of a self-similar process
$X(u,\oo)$ with self-similarity parameter $\alpha>0$ to the interval
$0\le u\le1$, and the weight functions $A_n$ and $B_n$ are
such that $A_n=B_n^{1/\alpha}$.
Define the indices $N(n)$ as $N(n)=\inf\{k\: B_k\ge 2^n\}$,
$n=0,1,\dots$. The pairs of sequences of random variables
$(\xi_n(\oo),\eta_n(\oo))$, $n=1,2,\dots$, satisfy Property~A if for
all $\e>0$ and $\delta>0$ there exists a sequence of random variables
$\tilde\xi_n(\oo)=\tilde\xi_n(\e,\delta,\oo)$, $n=1,2,\dots$, whose
(joint) distribution agrees with the (joint) distribution of the
sequence $\xi_n(\oo)$, $n=1,2,\dots$, and the partial sums $\tilde
S_n(\oo)=\summ_{k=1}^n\tilde\xi_k(\oo)$ and
$T_n(\oo)=\summ_{k=1}^n\eta_k(\oo)$ satisfy the following relation:
$$
\limsup_{n\to\infty}\frac1n\sum_{k=1}^{N(n)}\log\frac
{B_{k+1}}{B_{k}} I\(\left\{\frac{\supp_{0\le
j\le k}|\tilde S_j(\oo)-T_j(\oo)|}{A_k}>\e\right\}\)< \delta \tag2.13
$$
for almost all $\oo\in\Omega$, where $I(A)$ denotes the indicator
function of the set $A$.}
\medskip\noindent
{\it Remark:}\/ Let us remark that the joint distribution of the random
variables $\xi_n(\oo)$, $n=1,2,\dots$, determines whether it satisfies
the almost sure invariance principle. It is not important how and
on which probability space these random variables are
constructed. This can be seen for instance by applying the following
``canonical representation" of the sequence $\xi_n(\oo)$,
$n=1,2,\dots$, on the probability space $(\Omega,\Cal A,P)$. Define the
space $(R^\infty,\Cal B^\infty,\bar \mu)$, where
$R^\infty=\{(x_1,x_2,\dots)\: x_j\in R,\;j=1,2,\dots\}$, $\Cal
B^\infty$ is the Borel $\sigma$-algebra on $R^\infty$, $\bar\mu(\bold
B)=P((\xi_1,\xi_2,\dots)\in \bold B)$ for $\bold B\in \Cal B^\infty$,
and define the random variables $\bar \xi_n(x_1,x_2,\dots)=x_n$,
$n=1,2,\dots$, on this space. Then the random variables $\bar\xi_n$ on
the space $(R^\infty,\Cal B^\infty,\bar\mu)$ have the same joint
distribution as the random variables $\xi_n(\oo)$, and these two
sequences satisfy the almost sure invariance principle simultaneously.
\medskip\noindent
{\bf Theorem 4.} {\it Let $\eta_n(\oo)$, $n=1,2,\dots$, be a
sequence
of random variables which satisfies the almost sure functional limit
theorem, and let a pair of sequences of random variables
$(\xi_n(\oo),\eta_n(\oo))$, $n=1,2,\dots$, satisfy Property~A. Then
the sequence of random variables $\xi_n(\oo)$, $n=1,2,\dots$, also
satisfies the almost sure functional limit theorem with the same weight
functions $A_n$ and $B_n$ and limit measure $\mu_0$ as the sequence of
random variables~$\eta_n(\oo)$.}
\medskip
We shall prove in Part II.\ of this work that Theorem~4 is applicable
in several interesting cases. We shall prove with the help of a Basic
Lemma formulated there that when partial sums of independent random
variables are considered, then an appropriate construction satisfies the
conditions of Theorem~4 under general conditions. In such a way it will
turn out that the necessary and sufficient conditions of limit theorems
for normalized partial sums of independent random variables are also
sufficient conditions for the almost sure functional limit theorem.
We shall prove still another result which states that a small
perturbation of the weight functions $B_n$ does not affect the validity
of the almost sure functional limit theorem. The reason to prove such a
result is the following. We have certain freedom in the choice of the
weight-functions $B_n$, and there are cases when no ``most natural
choice" of the weight functions exists. We want to show that
different natural choices yield equivalent results. Let us
remark that a modification of the weight-functions $B_n$ also implies a
modification of the random broken lines $S_n(t,\oo)$ appearing in the
definition of the almost sure functional limit theorem.
\medskip\noindent
{\bf Theorem 5.} {\it Let a sequence of random variables
$\xi_n(\oo)$, $n=1,2,\dots$, satisfy the almost sure functional limit
theorem with some limit measure $\mu_0$ and weight functions $B_n$,
$A_n=B_n^{1/\alpha}$ with some $\alpha>0$, $n=0,1,\dots$, which
satisfies relation (2.11). Let us also assume that a process
$X(\cdot,\oo)$ in the space $D([0,1])$ whose distribution is the limit
measure $\mu_0$ satisfies condition (2.5). Let $\bar B_n$,
$n=0,1,\dots$, $\bar B_0=1$, be another monotone increasing sequence
such that $\limm_{n\to\infty}\dfrac{\bar B_n}{B_n}=1$. Put $\bar
A_n=\bar B_n^{1/\alpha}$. Then the sequence of random variables
$\xi_n(\oo)$ also satisfies the almost sure functional limit theorem
with the limit measure $\mu_0$ and weight functions $\bar B_n$ and~$\bar
A_n$.}\medskip
We shall prove Theorem 5 with the help of the following
Theorem~5A.\plainfootnote{*} {\small In the first version of this paper
the proof of Theorem 5 was incomplete. Unfortunately, I have observed
this only after the appearance of the paper. This problem is settled in
this version by the insertion and proof of Theorem~5A.}
\medskip\noindent
{\bf Theorem 5A.} {\it Let the conditions of Theorem 5 be satisfied.
Define the partial sums $S_n(\oo)=\summ_{k=1}^n\xi_k(\oo)$,
$n=1,2,\dots$, and the random broken lines $S_k(s,\oo)$ and $\bar
S_k(s,\oo)$, $0\le s\le 1$, $k=1,2,\dots$, by formula (2.11) with the
help of the constants $B_n$, $A_n=B_n^{1/\alpha}$ and $\bar B_n$, $\bar
A_n=\bar B_n^{1/\alpha}$ respectively. Let us also define the random
measures $\hat\mu_N(\oo)$, $N=1,2,\dots$, on the product space
$D([0,1])\times D([0,1])$ for all $\oo\in\Omega$ by the formula
$\hat\mu_N(\oo)(S_k(\cdot,\oo),\bar S_k(\oo))=\dfrac1{\log
\dfrac{B_N}{B_1}}\log\dfrac{B_{k+1}}{B_k}$, $1\le k\le N$.
For almost all $\oo\in\Omega$ and all $\delta>0$ the relation
$\limm_{N\to\infty}\hat\mu_N(\oo)\{(x,y)\:x,y\in D([0,1]),\,d(x,y)
\ge\delta\}=0$ holds, where $d(\cdot,\cdot)$ is the (complete) metric
introduced to define the topology in the space $D([0,1])$.}
\medskip\noindent
{\it Remark:} Actually the proof of Theorem ~5 yields a little bit more
than the result formulated there. It shows that under the conditions of
Theorem~5 the sequence of probability measures defined by formulas
(2.11) and (2.12) have the same weak limit for almost all $\oo\in\Omega$
as the original one if the random broken lines $S_k(s,\oo)$ are
replaced by $\bar S_k(s,\oo)$ or the weight functions
$\dfrac1{\log\dfrac{B_N}{B_1}}\log\dfrac{B_{k+1}}{B_k}$ are replaced by
$\dfrac1{\log\dfrac{\bar B_N}{\bar B_1}}\log\dfrac{\bar B_{k+1}}
{\bar B_k}$ in formula (2.12) or if both replacements are made.
Moreover, these statements hold if the condition
$\limm_{n\to\infty}\dfrac{\bar B_n}{B_n}=1$ in Theorem~5 is replaced by
the weaker condition $\bar B_n=B_nL(B_n)$, where $L(\cdot)$ is a slowly
varying function at infinity.
\beginsection 3. Proof of the results
{\it Proof of Theorem 1.} We can write
$$
Z(t+T,\oo)=\frac{X(e^{t+T},\oo)}{e^{(t+T)/\alpha}}\overset \Delta\to=
\frac{X(e^{t},\oo)}{e^{(t+T)/\alpha}e^{-T/\alpha}}=
\frac{X(e^{t},\oo)}{e^{t/\alpha}}=Z(t,\oo)
$$
for all $-\infty0$ there is a compact set $\bold K=\bold K(\e)$ on the metric space
such that $\mu_T(\bold K)\ge1-\e$ for all measures $\mu_T$. Both spaces
$C([0,1])$ and $D(0,1])$ can be endowed with a metric which turns them
to a separable complete metric space. (See e.g. Theorems 6.1 and 6.2,
14.1) in Billingsley's book [1].) Because of these results the following
statement has to be proved. For almost all $\oo\in\Omega$ and all $\e>0$
there exists a compact set $\bold K=\bold K(\e,\oo)$ in the space
$C([0,1])$ or $D([0,1])$ such that $\mu_T(\oo)(\bold K)\ge1-\e$
for all $T\ge1$. In the proof we shall apply formula (3.2) which is
valid for all bounded and measurable functionals $\Cal F$ and some
classical results which describe the compact sets in $C([0,1])$ and
$D([0,1])$. These results can be found for instance in the book of
Billingsley~[1]. (Theorem 8.2 gives a description of compact sets in
$C([0,1])$ and Theorem 14.4 a description of compact sets in
$D([0,1])$.)
Let us first consider the case when the distribution of the processes
$X_T(\cdot,\oo)$ defined in formula (2.4) are in the $C([0,1])$ space.
We shall prove that for almost all $\oo\in \Omega$ and all $\e>0$ and
$\eta>0$ there exist some numbers $K=K(\e,\oo)$ and
$\delta=\delta(\e,\eta,\oo)>0$ such that
$$
\align
&\mu_T(\oo)\(x\in C([0,1])\:\supp_{0\le u\le1}|x(u)|\ge K\)\le \e,\\
\intertext{and \hskip14truecm{(3.3)}}
&\mu_T(\oo)\(x\in C([0,1])\: |w_x(\delta)|\ge \eta\)\le \e,
\endalign
$$
for all $T\ge1$, where $w_x(\delta)=\supp_{|t-s|\le\delta}|x(t)-x(s)|$
for a function $x\in C([0,1])$. First we show that relation (3.3)
implies that for almost all $\oo\in\Omega$ and all $T\ge1$ and $\e>0$
there exists a compact set $\bold K(\e)=\bold K(\e,\oo)\subset
C([0,1])$ for which $\mu_T(\oo)(\bold K(\e))\ge 1-\e$. Indeed, let us
fix some $\e>0$, and consider the sets
$$
\bold J_0=\(x\in C([0,1])\:\supp_{0\le u\le1}|x(u)|> K\)
$$
and
$$
\bold J_n=\(x\in C([0,1])\: |w_x(\delta_n)|> 2^{-n}\e\),\quad n=1,2,\dots
$$
with such constants $K=K(\e,\oo)$ and $\delta_n=\delta_n(\e,\oo)$ for
which $\mu_T(\oo)(\bold J_n)\le\e 2^{-n-1}$, $n=0,1,\dots$, $T\ge1$.
Such sets $\bold J_n$ really exist because of relation (3.3). (The
numbers $K$ and $\delta_n$ in the definition of the sets $\bold J_n$
and thus the sets $\bold J_n$ may depend on $\oo$.) Define the set
$\bold K(\e)=\bigcapp_{n=0}^\infty\bar{\bold J}_n$, where $\bar{\bold
J}$ is the complement of the set $\bold J$. Then $\bold K(\e)$ is a
compact set in $C([0,1])$, and for almost all $\oo$ and $T\ge1$
$\mu_T(\oo)(\bold K(\e))\ge1-\e$.
Applying this result for all $\e_n=2^{-n}$,
$n=1,2,\dots$, we get a set of $\bar\Omega$ of probability one, such
that for all $\oo\in\bar\Omega$, $T\ge1$ and $\e>0$ there exists a
compact set $\bold K(\e)=\bold K(\e,\oo)$ such that $\mu_T(\oo)(\bold
K(\e))\ge1-\e$. In such a way we reduced the proof of Lemma~A in the
case of continuous trajectories $X(\cdot,\oo)$ to the proof of relation
(3.3).
To prove formula (3.3) we shall apply relation (3.2) with appropriate
functionals $\Cal F_1$ and $\Cal F_2$ on the space $C([0,1])$. Put
$$
\align
\Cal F_1(x)&=\Cal F_{1,K}(x)=I\(\supp_{0\le u\le 1}|x(u)|\ge K\) \\
\intertext{and}
\Cal F_2(x)&=\Cal F_{2,\delta,\eta}(x)=I\(\supp_{s,t\in[0,1]\:
|t-s|\le\delta} |x(s)-x(t)|\ge \eta\)
\endalign
$$
with appropriate constants $K>0$, $\eta>0$ and $\delta>0$. For fixed
$\e>0$ and $\eta>0$ the constants $K=K(\e)>0$ and
$\delta=\delta(\e,\eta)>0$ can be chosen in such a way that $E\Cal
F_1(X_1(\cdot,\oo))<\e^2$ and $E\Cal F_2(X_1(\cdot,\oo))<\e^2$. Then,
because of formula (3.2) for almost all $\oo\in\Omega$ there exists
such a threshold $T_0=T_0(\oo)$ for which $\intl\Cal
F_i(x)\,d\mu_T(\oo)(x)\le\e$ for all $T\ge T_0(\oo)$ and $i=1,2$. Since
$\Cal F_i(x)=0$ or $\Cal F_i(x)=1$, $i=1,2$, this relation implies that
$\mu_T(\oo)(x\:\Cal F_i(x)\neq0)\le\e$, for $T\ge T_0(\oo)$, $i=1,2$.
This means that relation (3.3) holds for $T\ge T_0(\oo)$. Furthermore,
since $X_{aT}(u,\oo)=a^{-1/\alpha}X_T\(au,\oo\)$ for all $00$ and sufficiently small $\delta>0$
which depend only on $T_0(\oo)$. Hence relation (3.3) holds not only
for $T\ge T_0(\oo)$ but also for all $T\ge1$ with a possible
modification of the constants $\delta(\e,\eta,\oo)$ and $K(\oo)$ in it.
The proof in the case when the processes $X_T(\cdot,\oo)$ defined in
{\rm (2.4)} take their values in the space $D([0,1])$ is similar, hence
we only indicate the necessary modifications. Because of the description
of compact sets in the space $D([0,1])$ found for instance in
Theorem~(14.4) in Billingsley's book [1]) we can reduce the proof of
Lemma~A in this case, by a natural modification of the argument
presented after the formulation of formula~(3.3), to the following
modified version of relation (3.3): For all $\e>0$ and $\eta>0$ there
exist some $K>0$ and $\delta>0$ such that
$$
\aligned
&\mu_T(\oo)\(x\in D([0,1])\:\supp_{0\le u\le1}|x(u)|\ge K\)\le \e,\\
&\mu_T(\oo)\(x\in D([0,1])\:|w''_x(\delta)|\ge \eta\)\le \e, \\
&\mu_T(\oo)(x\in D([0,1])\:w_x[0,\delta)\ge\eta)\le\e \\
&\mu_T(\oo)(x\in D([0,1])\:w_x[1-\delta,1)\ge \eta)\le\e
\endaligned
\tag$3.3'$
$$
for all $T\ge1$, where
$$
w_x''(\delta)=\supp_{0\le t_1\le
t\le t_2, |t_2-t_1|\le\delta}\min\{|x(t)-x(t_1)|,|x(t_2)-x(t)|\},
$$
and
$w_x[a,b)=\supp_{a\le s,t__**0$ and
$\e>0$ there exists a set $\Omega_0=\Omega_0(\e_0,\e)\subset\Omega$
and a compact set $\bold K=\bold K(\e_0,\e)$ in $C([0,1])$ or
$D([0,1])$ such that $P(\Omega_0)\ge1-\e_0$ and $\mu_T(\oo)(\bold
K)\ge1-\e$ for all $\oo\in\Omega_0$ and $T\ge1$. This can be deduced
from formulas (3.3) in the space $C([0,1])$ and from formula $(3.3')$
in the space $D([0,1])$ by an argument similar to the proof of the
compactness of the measures $\mu_T(\oo)$ by means of these relations.
Thus for instance in the space $C([0,1])$ we define the sets $\bold
J_n$, $n=1,2,\dots$, and $\bold K=\bold K(\e)$ similarly to the
definition given after formula (3.3) with the only difference that in
this case the numbers $K$ and $\delta_n$ appearing in the definition of
the sets $\bold J_n$ are chosen independently of $\oo$ in such a way
that $P(\{\oo\:\mu_T(\oo)(\bold J_n)\le\e2^{-n-1} \text{ for all }
T\ge1\})\ge1-\e_02^{-n-1}$. The argument in the case of the $D([0,1])$
space with the help of relation $(3.3')$ is similar.
For a large number $L>0$ let $\bold F(L)$ denote the class of continuous
and bounded functionals $\Cal F$ on the space $C([0,1])$ or $D([0,1])$
such that $|\Cal F(x)|\le L$ for all $x\in C([0,1])$ or $x\in D([0,1])$.
Fix an $\e_0>0$ and $\e>0$, and choose a set $\Omega_0\subset\Omega$
and a compact set $\bold K=\bold K(\e_0,\e, L)$ in such a way that
$P(\Omega_0)\ge1-\e_0$ and $\mu_T(\oo)(\bold K)\ge1-\dfrac\e L$ for
all $\oo\in\Omega_0$ and $T\ge1$. Fix two small numbers $\eta>0$
and $\delta>0$, and let the set $\bold F(L,\e_0,\e,\eta,\delta)\subset
\bold F(L)$ consist of those functionals $\Cal F\in\bold F(L)$ for
which $\supp_{x,y\in\bold K,\,\rho(x,y)\le \delta}|\Cal F(x)-\Cal
F(y)|\le\eta$. For all $\delta>0$ fix a finite $\delta$-net in the
compact set $\bold K$ corresponding to it, i.e.\ a finite set $\bold
J_\delta= \{x_1,\dots,x_r\}\subset \bold K$ such that for all $x\in
\bold K$ $\minn_{1\le s\le r}\rho(x,x_s)\le \delta$. Such a
$\delta$-net really exists because of the compactness of the set
$\bold K$.
Consider the above fixed numbers $\e_0>0$, $\e>0$ and $L>0$, together
with the sets $\Omega_0$ and $\bold K$ corresponding to them. First we
show that there exists an $\Omega_0'\subset \Omega_0$ such that
$P(\Omega_0\setminus\Omega_0')=0$, and
$$
\aligned
\limsup_{T\to\infty}&\left|\intl \Cal F(x)\mu_T(\oo)(\,dx)- \intl \Cal
F(x)\mu_0(\,dx)\right|<\e\\
&\qquad \text{for all }\Cal F\in \bold F(L)
\quad \text{and }\oo\in\Omega'_0.
\endaligned\tag3.4
$$
To prove relation (3.4) let us first observe that because of the uniform
continuity of the functionals $\Cal F\in \bold F_L$ on the compact set
$\bold K$ the relation
$$
\bigcupp_{n=1}^\infty\bold F\(L,\e_0,\e,\eta,\frac1n\)=\bold F(L)
\tag3.5
$$
holds for all fixed $\e_0>0$, $\e>0$, $\eta>0$ and $L>0$.
Put $\delta=\frac1n$, consider the $\frac1n$-net $\bold
J_{1/n}=\{x_1,\dots,x_r\}$ corresponding to it, and make a partition of
the set $\bold F\(L,\e_0,\e,\eta,\frac1n\)$ into subclasses
$\bold F\(L,\e_0,\e,\eta,\frac 1n, j(1),\dots,j(r)\)$
with integers
$|j(s)|\le (L+1)\eta^{-1}$, $s=1,\dots,r$, which consist of those
functionals $\Cal F\in\bold F\(L,\e_0,\e,\eta,\frac1n\)$ for which
$\Cal F(x_s)\in [j_s\eta,(j_s+1)\eta)$, $s=1,\dots,r$. If $\Cal F_1$ and
$\Cal F_2$ belong to the same subclass $\bold F(L,\e_0,\e,\eta, \frac
1n, j(1),\dots,j(r))$, then $|\Cal F_1(x)-\Cal F_2(x)|<2\eta$ for all
$x\in \bold K$ because of the module of continuity of these functionals
on the set $\bold K$, and because of the relation $\mu_T(\oo)(\bold
K)\ge 1-\dfrac\e L$ for all $\oo\in \Omega_0$,
$\left|\intl \Cal F_1(x)\mu_T(\oo)(\,dx)-\Cal F_2(x)\mu_T(\oo)(\,dx)
\right|<\e+2\eta$.
Let us choose an arbitrary functional $\Cal F$ from all non-empty sets
$$
\bold F\(L,\e_0,\e,\eta,\frac 1n, j(1),\dots,j(r)\).
$$
We get by applying formula (3.2) for these functionals $\Cal F$ and
the previous estimation a weakened version of relation (3.4) on a set
$\oo\in\Omega''_0(n)\subset \Omega_0$ such that
$P(\Omega_0\setminus\Omega''_0(n))=0$, where $\bold F(L)$ is replaced
by $\bold F\(L,\e_0,\e.\eta,\frac1n\)$, and the upper bound $\e$ by
$\e+2\eta$. Then we get, by applying this relation for all
$n=1,2,\dots$ together with relation (3.5) the weakened version
of (3.4) for all $\oo\in\bigcapp_{n=1}^\infty\Omega''_0(n)$ and $\Cal
F\in \bold F(L)$ with upper bound $\e+2\eta$ instead of $\e$. Finally,
we get formula (3.4) in its original form by letting $\eta\to0$.
It is not difficult to see that relation (3.4) implies the weak
convergence $\mu_T(\oo)$ to $\mu_0$ for almost all $\oo\in\Omega$.
Indeed, let us fix a number $L>0$ and $\e>0$. Then we get, by
applying relation (3.4) for all $\e_0(n)=n^{-1}$, $n=1,2,\dots$ that
there exists a set $\Omega_0(n)$, $P(\Omega_0(n))=1-\dfrac1n$, such
that relation (3.4) holds for all $\oo\in\Omega_0(n)$. This implies
that relation (3.4) holds for all $\oo\in\bar\Omega
=\bigcupp_{n=1}^\infty\Omega_0(n)$, i.e.\ on a set of probability~1.
Then, since relation (3.4) holds for all $L>0$ and $\e>0$ with
probability~1 we get by letting $L\to\infty$ and $\e\to0$ in this
relation that the sequences of measures $\mu_T(\oo)$ converge weakly to
the measure $\mu_0$ for almost all $\oo\in\Omega$.
To complete the proof of Theorem~1 still we have to show that in the
case of a Wiener or a stable process the generalized
Ornstein--Uhlenbeck process corresponding to it is ergodic. This
follows from a natural modification of the zero--one law for sums of
independent identically distributed random variables to processes with
independent and stationary increments which can be found for instance
in Feller's book [4], Chapter~4, Section~7, Theorem~3. The continuous
time version of this result which can be proved similarly, also holds.
It states that if $X(t)$, $t\ge0$, is a stable process with some
parameter~$\alpha$, $0<\alpha\le2$, and a set $\bold A$ is measurable
with respect to the (tail) $\sigma$-algebra $\Cal F$ which is the
intersection $\Cal F=\bigcapp_{T>0}\Cal F_T$, where $\Cal F_T=\sigma
\{X(t,\cdot)\: t\ge T\}$, then $\bold A$ has probability zero or
one. The same result holds if the set $\bold A$ is measurable with
respect to the $\sigma$-algebra $\bigcapp_{T>0}\Cal F'_T$, where
$\Cal F'_T=\sigma\{X(t,\cdot)\:t\le T\}$. (This result follows
for instance from the observation that $t^{-2/\alpha}X\(\frac1t,\oo\)$
is also a stable process. These relations are equivalent to the
statement that the generalized Ornstein--Uhlenbeck process $Z(t)$
corresponding to this stable process has trivial $\sigma$-algebra at
infinity and minus infinity, i.e.\ all sets which are measurable with
respect to the $\sigma$-algebra generated by the random variables $t\ge
T$ (or $t\le T$) for all $-\infty0$ and
$\delta>0$ as
$$
\Cal F_{\e,\delta}(x)=I\(\supp_{1-\e\le
s,t\le1}\rho(x_s(\cdot),x_t(\cdot)) \ge \delta\),
$$
where the function $x_t$ is defined in $(2.4')$, and
$\rho(\cdot,\cdot)$ is the metric introduced in Section~2. We claim
that under the conditions of Theorem~2
$$
\limm_{\e\to0}E\Cal F_{\e,\delta}(X_1(\cdot,\oo))=0 \tag3.6
$$
for all $\delta>0$.
Let us also observe that by relation (3.2)
$$
\lim_{T\to\infty}\mu_T(\oo)\(\sup_{1-\e\le s,t\le1}\rho(x_s,x_t)
>\delta\)=\lim_{T\to\infty}\intl \Cal
F_{\e,\delta}(x)\,d\mu_T(\oo)(x)
=E \Cal F_{\e,\delta}(X_1(\cdot,\oo))
$$
for all $\e>0$ and $\delta>0$ and almost all $\oo$, where the function
$x_t$ was defined in formula $(2.4')$. Then we get relation (2.6) with
the help of formula (3.6), by letting $\e\to0$ in the last formula.
Hence to prove relation (2.6) it is enough to prove formula (3.6).
If $X_1(\cdot,\oo)\in C([0,1])$, then this relation follows from the
observation that for all $\eta>0$ there is a compact set $\bold K_\eta$
in $C([0,1])$ such that $P(X_1(\cdot,\oo)\in \bold K_\eta)\ge1-\eta$,
and for all $\delta>0$ there exists an $\e=\e(\eta)>0$ such that
$|x(u)-x(v)|< \delta$ if $x\in \bold K_\eta$, and $|u-v|\le \e$. There
is also a constant $L>0$ such that $\supp_{x\in\bold K_\eta} |x(u)|\le
L$. Since these relations hold for all $\delta>0$ and appropriate $L>0$
they imply that $\limm_{\e\to0}\supp_{x\in \bold K_\eta, 1-\e\le t\le 1}
\rho(x_t,x)=0$. This means that for sufficiently small $\e>0$
$\Cal F_{\e,\delta}(X_1(\cdot,\oo))=0$ if $X_1(\cdot,\oo)\in \bold
K_\eta$, i.e.\ in the case when an event of probability greater than
$1-\eta$ occurs. Hence relation (3.6) holds in this case. The situation
in the space $D([0,1])$ is more sophisticated. In this case formula
(2.5) also has to be applied.
Since all functions $x(t)$ in the space $D([0,1])$ have a limit as
$t\to1-0$ it follows from relation (2.5) that for all $\delta>0$
$$
P\(\lim_{\e\to0}\supp_{1-\e\le t\le1}|X(t,\oo)-X(1,\oo)|
\ge\dfrac\delta2\)=0.
$$
Hence there is a set $\bold K=\bold K_\eta$ in the space $D([0,1])$
such that $P(X_1(\cdot,\oo)\in \bold K)\ge1-\eta$, the closure of the
set $\bold K$ is compact, and for all $x\in\bold K$
$\limm_{\e\to0}\supp_{1-\e\le t\le1}|x_t-x|<\dfrac\delta2$, where the
function $x_t$ was defined in $(2.4')$. There is a finite
$\dfrac\delta5$--net in $\bold K$, i.e.\ a
finite set $\bold J=\{x^{(1)},\dots,x^{(s)}\}$, $x^{(r)}\in \bold K$,
$r=1,\dots,s$, in such a way that for all $x\in \bold K$ there is some
$x^{(r)}\in \bold J$ such that $\rho(x,x^{(r)})\le\dfrac \delta5$. Then
to prove formula (2.6) it is enough to show that for all $x^{(r)}\in
\bold J$ there is some $\bar\e>0$ such that $\rho(x_t^{(r)},x^{(r)})
\le\dfrac\delta4$ for all $1-\bar\e\le t\le1$. Indeed, if this
statement holds, then for arbitrary $x\in \bold K$ there is some
$x^{(r)}\in\bold J$ such that $\rho(x,x^{(r)})\le\dfrac\delta5$.
Then $\rho(x_s,x_t)\le\rho(x_s,x^{(r)}_s)+\rho(x_t,x^{(r)}_t)
+\rho(x_s^{(r)},x_t^{(r)})$. Let us also observe that because of the
definition of the functions $x_t$ for sufficiently small $\bar\e>0$
for all $x\in D([0,1])$, $1-\bar\e\le t\le1$ and $x^{(r)}\in \bold J$
the inequality $\rho(x_t,x^{(r)}_t)\le\dfrac54\rho(x,x^{(r)})$ holds,
and $\rho(x_s^{(r)},x_t^{(r)})\le \rho(x_s^{(r)},x^{(r)})
+\rho(x_t^{(r)},x^{(r)})$. The above inequalities imply that
$\rho(x_s,x_t)\le\delta$ for $1-\bar\e\le s,t\le1$ if $x\in\bold K$.
Hence $\Cal F_{\e,\delta}(X_1(\cdot,\oo))=0$ with $\e=\bar\e$ if
$X_1(\cdot,\oo)\in \bold K$. Then formula (3.6) follows from the
relation $P(X_1(\cdot,\oo)\in \bold K)\ge1-\eta$.
Thus to complete the proof of formula (2.6) it is enough to show that
for an arbitrary function $x\in D([0,1])$ such that
$\limm_{u\to1-0}|x(u)-x(1)|<\dfrac\delta2$ the relation
$\limm_{\e\to0}\rho(x_t,x)<\dfrac\delta2$ holds. (This relation means
in particular that the limit exists.) To prove this relation let us
define for all $\dfrac12\le t<1$ the mapping $\lambda_t(u)$ of the
interval $[0,1]$ into itself as $\lambda_t(u)=tu$ for $0\le u\le
t^*(t)$ with $t^*(t)=1-\sqrt{1-t}$, and define $\lambda_t(u)$ in the
remaining interval $(t^*(t),1]$ also linearly, i.e.\ let
$\lambda_t(u)=(\sqrt{1-t}+t)u+1-t-\sqrt{1-t}$ for $t^*(t)\le
u\le1$. Then $\limm_{t\to1}\supp_{u\neq v}\log\left|\dfrac
{\lambda_t(u)-\lambda_t(v)}{u-v}\right|=0$. Because of the definition
of the metric $\rho=d_0$ it is enough to show that
$$
\limm_{t\to1}\supp_{0\le u\le1}|x_t(u)-x(\lambda_t(u))|
=\limm_{u\to1}|x(u)-x(1)|<\dfrac\delta2.
$$
It is known that for an $x\in D([0,1])$ function $\supp_{0\le u\le
1}|x(u)|<\infty$ (see e.g.\ Billingsley's book [1]). Hence
$$
\align
\supp_{0\le u\le t^*(t)}|x_t(u)-x(\lambda_t(u))|
&\le(t^{-1/\alpha}-1)\supp_{0\le u\le1}|x(u)|\\
&\le\const(t^{-1/\alpha}-1)\to0 \quad\text{if }t\to1-0.
\endalign
$$
Similarly, since a function $x\in D([0,1])$ has a right-hand side limit
in the point 1, $\supp_{t^*(t)\le u<1}|x_t(u)-x(\lambda_t(u))|\to0$ as
$t\to1-0$. Finally in the point $u=1$ $\lambda_t(1)=1$, and
$\limm_{t\to1-0}\left|x_t(1)-x(\lambda_t(1))\right|=
\left|x(1)-\limm_{t\to1-0}x(t)\right|<\dfrac\delta2$. These relations
imply that $\limm_{t\to1-0}\rho(x_t,x)=\limm_{t\to1-0}|x(t)-x(1)|
<\frac\delta2$. Theorem 2 is proved.
\medskip\noindent
{\it Proof of Lemma 1.}\/ We have to prove that for arbitrary
$\delta>0$
$$
\lim_{\e\to0}P\(\sup_{1-\e\le t\le1}|X(t,\oo)-X(1,\oo)|>\delta\)=0.
$$
Because of the stationary increment and self-similarity property of the
process $X(t,\oo)$ with parameter $\alpha>0$ yields that
$$
\align
P&\(\sup_{1-\e\le t\le1}|X(t,\oo)-X(1,\oo)|>\delta\)
=P\(\sup_{0\le t\le\e}|X(t,\oo)-X(\e,\oo)|>\delta\)\\
&\qquad=P\(\sup_{0\le t\le1}|X(t,\oo)-X(1,\oo)|>\delta\e^{-1/\alpha}\).
\endalign
$$
Then tending with $\e\to0$ we get that $\delta\e^{-1/\alpha}\to\infty$,
and the required property holds.
To prove Theorem 3 first we formulate and prove the following technical
Lemma:
\medskip\noindent
{\bf Lemma B.} {\it Let $(M,\Cal M, \rho)$ be a separable, complete
metric space such that $\Cal M$ is the $\sigma$-algebra generated by
the open sets of this space. Let two sequences of probability
measures $\mu_N$ and $\bar \mu_N$, $N=1,2,\dots$, be given on the space
$(M,\Cal M,\rho)$ such that the measures $\mu_N$ weakly converge to a
probability measure $\mu_0$ on $(M,\Cal M, \rho)$ as $N\to\infty$, and
$$
\liminf_{N\to\infty}\(\bar\mu_N(\bold F^\e)-\mu_N(\bold F)\)\ge0\quad
\text{for all closed sets } \bold F\in \Cal M \text{ and } \e>0,
\tag3.7
$$
where $\bold A^\e=\{x\:\rho(x,\bold A)<\e\}$
denotes the $\e$-neighborhood of a set $\bold A\in\Cal M$. Then the
measures $\bar\mu_N$ converge weakly to the same limit measure $\mu_0$
as $N\to\infty$. Moreover, condition (3.7) can be slightly weakened.
It is enough to assume that it holds for all compact sets $\bold
K\in\Cal M$ and $\e>0$.}
\medskip\noindent
{\it Proof of Lemma B.}\/ The weak convergence of the measures
$\bar\mu_N$ to $\mu_0$ as $N\to\infty$ is equivalent to the relation
$\liminff_{N\to\infty}\bar\mu_N(\bold G)\ge \mu_0(\bold G)$ for all open
sets $\bold G\in\Cal M$. For all open sets $\bold G\in\Cal M$ and $\e>0$
there exists a compact set $\bold K=\bold K_\e\in \Cal M$ such that
$\bold K\subset \bold G$ and $\mu_0(\bold K)\ge\mu_0(\bold G)-\e$.
Then there exists some $\eta>0$ such that also the $\eta$-neighborhood
of $\bold K$ satisfies the relation $\bold K^\eta\subset\bold G$.
Consider the $\eta/2$ neighborhood $\bold K^{\eta/2}$ of the set
$\bold K$. Since $\bold G$ contains the $\eta/2$ neighborhood of the
closure of $\bold K^{\eta/2}$, and the measures $\mu_N$ converge weakly
to the measure $\mu_0$ as $N\to\infty$ we can write with the help of
relation (3.7) that $\liminff_{N\to\infty}\bar\mu_N(\bold G)\ge
\liminff_{N\to\infty}\mu_N(\bold K^{\eta/2})\ge \mu_0(\bold
K^{\eta/2})\ge\mu_0(\bold G)-\e$. Since the last relation
holds for all $\e>0$ and open sets $\bold G$, it implies the
convergence of the measures $\bar\mu_N$ to $\mu_0$ as $N\to\infty$.
To complete the proof of Lemma B let us observe that because of the
compactness (convergence) of the measures $\mu_N$ in the weak
convergence topology for all $\e>0$ there is a compact set $\bold K\in
\Cal M$ such that $\mu_N(\bold K)>1-\e$ for all $N=1,2,\dots$. Then
for a closed set $\bold F\in\Cal M$ the set $\bold F\cap\bold K$ is
also compact, and $\liminff_{N\to\infty}\(\bar\mu_N(\bold F^\e)-
\mu_N(\bold F)\)\ge \liminff_{N\to\infty}\(\bar\mu_N((\bold F\cap\bold
K)^\e)-\mu_N(\bold F\cap\bold K)\)-\e\ge-\e$. Since this relation holds
for all $\e>0$, it is enough to assume relation~(3.7) for compact
sets~$\bold K$. \medskip
Now we introduce the notion of good coupling we shall use later and
formulate a simple consequence of Lemma~B.
\medskip\noindent
{\bf Definition of good coupling:} {\it Let two sequences of probability
measures $\mu_N$ and $\bar\mu_N$, $N=1,2,\dots$, be given on a separable
complete metric space $(M,\Cal M,\rho)$, where $\Cal M$ denotes the
$\sigma$-algebra generated by the topology induced by the metric
$\rho$. These two sequences of measures have a good coupling if for all
$\e>0$ and $\delta>0$ there is a sequence of probability measures
$P^{\e,\delta}_N$, $N=1,2,\dots$, on the product space $(M\times M,\Cal
M\times\Cal M,\bar\rho)$, $\bar\rho((x_1,y_1),(x_2,y_2))=
\rho(x_1,x_2)+\rho(y_1,y_2)$ which satisfies the following properties.
\item{i.)} The marginal distributions of $P^{\e,\delta}_N$ are $\mu_N$
and $\bar\mu_N$, i.e. $P^{\e,\delta}_N(\bold A\times M)=\mu_N(\bold A)$
and $P^{\e,\delta}_N(M\times \bold A)=\bar\mu_N(\bold A)$ for all
$\bold A\in\Cal M$, and $n=1,2,\dots$.
\item {ii.)} $\limsupp_{N\to\infty}P^{\e,\delta}_N(\{(x,y)\:
\rho(x,y)>\e\})\le\delta$.}
\medskip\noindent
{\bf Corollary of Lemma B.} {\it If two sequences of probability
measures $\mu_N$ and $\bar\mu_N$, $N=1,2,\dots$, on a complete
separable metric space $(M,\Cal M,\rho)$ have a good coupling, and the
sequence of measures $\mu_N$ converge weakly to a probability measure
$\mu_0$, then the measures $\bar\mu_N$ converge weakly to the same
measure $\mu_0$.}
\medskip\noindent
{\it Proof of the Corollary.} Fix an $\e>0$. For all $\delta>0$ we can
write
$$
\liminf_{N\to\infty}\(\bar\mu_N(\bold F^\e)-\mu_N(\bold F)\)\ge
-\limsupp_{N\to\infty} P^{\e,\delta}_N(\{(x,y)\:\rho(x,y)>\e\})\ge
-\delta.
$$
We get the statement of the Corollary by letting $\delta\to0$.
\medskip\noindent
{\it Proof of Theorem 3.}\/ We shall prove the weak convergence
of the
measures $\hat\mu_N(\oo)$ for almost all $\oo$ with the help of Lemma~B
with the choice of $\mu_{B_N}(\oo)$ as $\mu_N$ and $\hat\mu_N(\oo)$ as
$\bar\mu_N$. Then (for almost all $\oo$) the measures $\mu_N$ converge
weakly to $\mu_0$, and it is enough to show that for almost all
$\oo\in\Omega$
$$
\aligned
\liminf_{N\to\infty}&\(\hat\mu_N(\oo)(\bold F^\e)-\mu_{B_N}(\oo)(\bold
F)\)\ge0\quad \text{for all closed sets }\\
&\qquad\bold F\subset D([0,1])\text{ or }\bold F \subset C([0,1])
\text{ and } \e\ge0.
\endaligned \tag3.8
$$
Let us recall that for arbitrary measurable set $\bold B\subset
D([0,1])$ (or $\bold B\subset C([0,1])$)
$$
\align
\mu_{B_N}(\oo)(\bold B)&=\bar \lambda_{B_N}
\{s\: s\in [1,B_N],\; X_s(\cdot,\oo)\in \bold B\}\\
\intertext{and}
\bar\mu_N(\oo)(\bold B)&=\bar \lambda_{B_N}
\{s\: \text{there is some } 1\le j< k_N \text{ such that} \\
&\qquad\qquad B_{j,N}\le s0$ and $\eta>0$ define the set
$$
\bold A(\e,\eta)=\left\{x\in D([0,1])\: \supp_{1-\eta**~~0$ and $\delta>0$ fix some $\eta=\eta(\oo,\e,\delta)>0$
and $N_0=N_0(\oo,\e,\delta)$ in such a way that $\mu_{B_N}(\oo)(\bold
A(\e,\eta))>1-\delta$ for $N\ge N_0$. By Theorem~2 such a choice of
$\eta$ and $N_0$ is possible for almost all $\oo\in\Omega$. Then we can
choose, since the numbers $B_{k,j}$ satisfy condition (2.7), some
number $j_0=j_0(\eta)$ and $N_1\ge N_0$ in such a way
that $\dfrac{B_{k+1,N}}{B_{k,N}}\le 1+\frac\eta2$, if $N\ge N_1$ and
$j_0\le k\e\right\}
$$
for all $\e>0$. Since the measures $\hat\mu_N$ are compact for
all $\eta>0$ there is
a compact set $\bold K=\bold K(\eta)\subset D([0,1])$ such that
$\hat\mu_N(\bold K)>1-\eta$ for all $N=1,2,\dots$, and formula (3.10)
can be reduced to the statement
$$
\limm_{N\to\infty}P_N(\oo)(\bold A_N(\e,\oo)\cap(\bold K\times
D([0,1])))=0 \tag3.11
$$
for arbitrary compact set $\bold K\subset D([0,1])$. Moreover, this
statement can be reduced to a slightly weaker statement. To formulate it
let us define for all $\eta>0$ and $N=1,2,\dots$ the number $\hat
\jmath(N)=\hat \jmath(N,\eta)$ as $\hat \jmath(N)=\max\{j\: \log
B_{j,N}\le \eta\log B_N\}$. Because of condition (2.7) imposed on the
numbers $B_{j,k}$ in Theorem~3 $\hat\jmath(N)\to\infty$ as
$N\to\infty$. Because of the definition of the measures
$\hat\mu_N(\oo)$ and the number $\hat \jmath(N)$ the inequality
$\hat\mu_N(\oo)\left\{\bigcupp_{j\:j\le\hat \jmath(N)}X_{B_{j,N}}
(\cdot,\oo)\right\}\le\eta$ holds. Define the set
$$
\align
\bold A_N^\eta(\e,\oo)&=\left\{(X_{B_j,N}(\cdot,\oo),\bar
X_{B_j,N}(\cdot,\oo))\: \hat \jmath(N,\eta)\le j\le k_N,\right. \\
&\qquad\qquad\qquad\qquad \left. \rho(X_{B_j,N}(\cdot,\oo),\bar
X_{B_j,N}(\cdot,\oo))>\e\right\}.
\endalign
$$
Then $\hat \mu_N(\oo)(\bold A_N(\e,\oo)\setminus\bold A^\eta_N(\e,\oo))
\le \eta$, and relation (3.11) can be reduced to the relation
$$
\limm_{N\to\infty}P_N(\oo)(\bold A_N^\eta(\e,\oo)\cap(\bold K\times
D([0,1])))=0 \tag$3.11'$
$$
by letting $\eta\to0$.
We claim that for an arbitrary compact set $\bold K\subset D([0,1])$,
$\e>0$ and $\eta>0$ there is some $N_0=N_0(\bold K,\e,\eta,\oo)$ such
that for all $N\ge N_0$ and $j\ge \hat \jmath(N)$ the relation
$X_{B_{j,N}}(\cdot,\oo)\in \bold K$ implies that
$\rho(X_{B_{j,N}}(\cdot,\oo),\bar X_{B_{j,N}}(\cdot,\oo))<\e$, hence
the set $ \bold A_N^\eta(\e,\oo)\cap (\bold K\times D([0,1]))$ is empty
for large enough~$N$. This statement clearly implies relation
$(3.11')$.
To prove this statement let us observe that the trajectory $\bar
X_{B_{j,N}}(\cdot,\oo)$ is obtained as a discretization
of the trajectory $X_{B_{j,N}}(\cdot,\oo)$ of the following type:
There is a partition $0=t_{j,0,N}0$ there exist some
$\hat\jmath_1=\hat\jmath_1(\eta)$, $\hat\jmath_2=\hat\jmath_2(\eta)$
and $N_0=N_0(\eta)$ in such a way that
$\dfrac{B_{l,N}}{B_{l-1,N}}\le1+\frac\eta2$ if $\hat \jmath_1\le l\le N$
and $N\ge N_0$, and $\eta B_{\hat \jmath_2,N}\ge
B_{\hat\jmath_1,N}$ if $N\ge N_0$. Then for all $N\ge j\ge \hat\jmath_2$
and $N\ge N_0$ $t_{j,l,N}-t_{j,l-1,N}\le \dfrac{B_{l,N}-B_{l-1,N}}
{B_{l,N}}\le\eta$ for $j\ge l\ge \hat\jmath_1$, and
$t_{j,l,N}-t_{j,l-1,N}\le\dfrac{B_{\hat\jmath_1,N}}{B_{\jmath_2,N}}
\le\eta$ if $1\le l\le \hat \jmath_1$. The width of the partitions
considered above
tends to zero if $\hat\jmath=\hat\jmath(N)\to\infty$, as we claimed.
Indeed, the previous calculations imply that it is less than $\eta$ for
$\hat\jmath\ge\hat\jmath_2(\eta)$.
We claim that this relation implies that
$$
\limm_{N\to\infty}\supp_{j\: j\ge\hat \jmath(N),\,X_{j,N}(\cdot,\oo)\in
\bold K} \rho(X_{j,N}(\cdot,\oo),\bar X_{j,N}(\cdot,\oo))=0
$$
for all compact sets~$\bold K\subset D([0,1])$, and this relation
implies formula $(3.11')$ and hence the second part of Theorem~3.
Let us define the following function $g(x,\delta)$ for $x\in D([0,1])$
and $\delta>0$:
$$
g(x,\delta)=\sup\Sb 0=t_00$ there is some
$\alpha=\alpha(\eta)>0$ and a partition $0=u_0\alpha$, and
$\supp_{1\le j\le r}\supp_{u_{j-1}\le s,t0$, it implies the first statement of Lemma~C.
To prove this statement let us consider the partition
$0=T_00$ and $\delta>0$ in the definition are the same
$\e$ and $\delta$ which appear in formula~(2.13).)
Then the marginal distributions of $P^{\e,\delta}_N(\oo)$ are the
distributions $\mu_N(\oo)$ and $\bar\mu_N(\oo)$ appearing in the
definition of the almost sure functional limit theorem. By the
Corollary of Lemma~B it is enough to prove that
$$
\limsup_{N\to\infty}P_N^{\e,\delta}(\oo)\{(x,y)\:\rho(x,y)>\e\}<\delta
$$
for almost all $\oo\in\Omega$. Since $\rho(x,y)\le d(x,y)$ with
$d(x,y)=\supp_{0\le u\le1}|x(u)-y(u)|$,
$$
P_N^{\e,\delta}(\oo)\{(x,y)\:\rho(x,y)>\e\}\le\frac2{\log B_N}
\sum_{k=1}^{N-1}\log\frac{B_{k+1}}{B_{k}} I(d(\tilde
S_k(\cdot,\oo),T_k(\cdot,\oo))>\e)
$$
for sufficiently large $N$. For a number $N$ choose the number
$\bar n=\bar n(N)$ such that $2^{\bar n-1}\le B_N<2^{\bar n}$. Then
$N\le N(\bar n)$, and $\log B_N\ge \bar n-1$. Hence
$$
P_N^{\e,\delta}(\oo)\{(x,y)\:\rho(x,y)>\e\}
\le\frac1{\bar n-1}\sum_{k=1}
^{N(\bar n)} \log\frac {B_{k+1}}{B_{k}} I\(\left\{\frac{\supp_{1\le
j\le k} |\tilde S_j(\oo)-T_j(\oo)|}{A_k}>\e\right\}\)
$$
with this $\bar n=\bar n(N)$. As $\bar n(N)$ tends to infinity as
$N\to\infty$ relation (2.13) implies that the $\limsup$ of the
right-hand side of the last expression is less than $\delta$ for almost
all $\oo$ as $N\to\infty$. Theorem~4 is proved.
\medskip\noindent
{\it Proof of Theorem 5A.}\/
Let us consider the partial sums $S_k(\oo)
=\summ_{j=1}^k\xi_j(\oo)$, $k=1,2,\dots$, and the random
polygons $S_n(s,\oo)$ and $\bar S_n(s,\oo)$, $n=1,2,\dots$, defined by
formula (2.11) with weight functions $B_n$, $A_n=B_n^{1/\alpha}$ and
$\bar B_n$, $\bar A_n=\bar B_n^{1/\alpha}$ respectively. Let us also
introduce the random polygons $\bar S'_n(\oo)$ defined with the help of
the partial sums $S_k(\oo)$ with the new weight functions $\bar B_n$
and the original sequence $A_n=B_n^{1/\alpha}$ by formulas (2.11).
We have to compare the distance $\rho(S_N(\cdot,\oo),\bar
S'_N(\cdot,\oo))\le \e$.
It is not difficult to show that $\limm_{N\to
\infty}d(S_N(\cdot,\oo),\bar S_N'(\cdot,\oo))=0$
under the conditions of Theorem~5, if the metric $\rho=d_0$ applied in
this paper is replaced by the following metric $d(\cdot,\cdot)$ in the
space $D([0,1])$: The relation $d(x,y)\le \e$, $x,y\in D([0,1])$,
holds, if there is a strictly monotone increasing continuous function
$\lambda(t)$ which is a homeomorphism of the interval $[0,1]$ into
itself, and $\supp_{0\le t\le 1}|\lambda(t)-t|\le \e$, $\supp_{0\le
t\le 1}|y(t)-x(\lambda(t))|\le \e$. The metric $d$ induces the same
topology as the metric $\rho=d_0$ in the space $D([0,1])$, but it has
the unpleasant property that the space $D([0,1])$ is not a complete
metric space with this metric. A detailed discussion about the relation
between the metrics $d(\cdot,\cdot)$ and $d_0(\cdot,\cdot)$ is
contained in the book of Billingsley~[1].
In the proof we have to overcome the following difficulty. The natural
transformation $\lambda(\cdot)$ for which $\bar
S_N(\lambda(\cdot,\oo))$ is close to $\bar S'_N(\cdot,\oo)$ is
the map which transforms the point $\frac{\bar B_k}{\bar B_N}$ to the
point $\frac{B_k}{B_N}$, and is linear between these points. This
transformation shows that for large $N$ the corresponding trajectories
are close in the $d(\cdot,\cdot)$ metric, but it supplies no good
estimate for the distance in the $d_0(\cdot,\cdot)$ metric.
We recall the following result from Billingsley's book~[1] (see Lemma~2
in Section~14): If $d(x,y)\le \delta^2$, $0< \delta\le 1/4$, then
$\rho(x,y)=d_0(x,y)\le 4\delta+w'_x(\delta)$, where the inequality
$w'_x(\delta)\le\e$ for a function $x\in D([0,1])$ means that there
exist some numbers $0=t_00$ and
the compact set $\bold K=\bold K(\e,\oo)\subset D([0,1])$ we have fixed
choose a number $0<\eta<1/4$ such that $5\eta<\delta/2$ and a number
$\bar\eta>0$ such that $w'_x(\bar\eta)<\eta$ if $x\in \bold K$. Then
there is an index $N_0=N_0(\eta,\bar\eta)$ such that
$d(S_N(\cdot,\oo),\bar S'_N(\cdot,\oo))\le \min(\eta^2,\bar\eta^2)$, if
$N\ge N_0$. The above relations imply that $\rho(S_N(\cdot,\oo),\bar
S_N(\cdot,\oo))\le4\min(\eta,\bar\eta)+w'_{S_N(\cdot,\oo)}(\bar\eta)\le
\delta/2$, if $N\ge N_0$ and $S_N(\cdot,\oo)\in\bold K$.
To complete the proof of Theorem~5A we compare the random broken
lines $\bar S_n'(\oo)$ and $\bar S_n(\oo)$. Observe that $\bar
S'_k(\cdot,\oo)=\dfrac{\bar A_k}{A_k} \bar S_k(\cdot,\oo)$, and
$\limm_{k\to\infty}\dfrac{\bar A_k}{A_k}=1$. On the other hand, given
the compact set $\bold K=\bold K(\e,\oo)$, there is a number
$K=K(\e,\oo)>0$ such that $\supp_{x\in\bold K}\supp_{0\le s\le
1}|x(s)|\le K$. These facts imply that there exists
some threshold index $N_1=N_1(\oo,\e)$ such that $\rho(\bar
S_N(\cdot,\oo),\bar S_N'(\oo))\le \delta/2$ if $N\ge N_1$.
The previous arguments imply that there is some index
$\bar N=\max(N_0,N_1)$ and a compact set $\bold K\in D([0,1])$
such that $\mu_N(\oo)(\bold K)\ge1-\e$, and $\rho(S_N(\cdot,\oo),
\bar S_N(\cdot,\oo))\le \delta$ if $N\ge N_1$ and $S_N(\cdot,\oo)\in
\bold K$. Since $\limm_{N\to\infty}\mu_N(S_k(\cdot,\oo))=0$ for all
fixed $k>0$, the $\mu_N(\oo)$ probability of the random broken lines
$S_n(\cdot,\oo)$ for which $\rho(S_n(\cdot,\oo),\bar S_n(\oo))\le
\delta$ is less than~$2\e$. Since this relation holds for all $\e>0$,
it implies Theorem~5A. \medskip\noindent
{\it Proof of Theorem 5.} First we prove the following statement. Let us
fix some $\delta>0$ and let $\bar{\bold K}$ be a compact set in the
space $D([0,1])$ which also satisfies the following property: There is
some $\eta_0>0$ such that
$$
\supp_{1-\eta_0\le u\le1}|x(u)-x(1)|\le
\frac\delta4\quad \text{for all }x\in \bold K. \tag3.13
$$
We claim that there exists a number $\eta=\eta(\delta,\eta_0,\bar {\bold
K})>0$ such that for all functions $x\in\bar {\bold K}$ and numbers
$1-\eta\le t\le 1$ the inequality $d(x,x_t)<\delta$ holds, where the
function $x_t$ is defined in formula ($2.4'$), and $d(\cdot,\cdot)$ is
the complete metric we introduced to define the topology in the space
$D([0,1])$.
To prove this statement let us first observe that because of the
compactness of the set $\bar{\bold K}$ there exists a number $K>0$
such that $\supp_{x\in\bar{\bold K}}\supp_{0\le u\le 1}|x(u)|\le K$.
Given a function $x(\cdot)\in D([0,1])$ and a number $00$ such that $d(x_t,\bar
x_t)0$ in such a way that $d(x,\bar x_t)\le \frac\delta2$ if
$x\in\bar{\bold K}$ and $1-\eta'\le t\le1$.
To prove this statement let us define for all $\frac12\le t<1$ the
mapping $\lambda_t(u)$ of the interval $[0,1]$ into itself as
$\lambda_t(u)=tu$ for $0\le u\le t^*(t)$ with $t^*(t)=1-\sqrt{1-t}$,
and define $\lambda_t(u)$ in the remaining interval $(t^*(t),1]$ also
linearly, i.e.\ let $\lambda_t(u)=(\sqrt{1-t}+t)u+1-t-\sqrt{1-t}$ for
$t^*(t)\le u\le1$. There is some $\eta_2>0$ such that $\supp_{u\neq
v}\log\left|\dfrac {\lambda_t(u)-\lambda_t(v)}{u-v}\right|\le
\dfrac\delta2$ if $1-\eta_2\le t\le1$. By recalling the definition of
the metric $d(\cdot,\cdot)$ we see that to complete the proof
of the statement we claimed to hold it is enough to show that there is
some $\eta_3>0$ such that for all $x\in \bar{\bold K}$ and
$1-\eta_3\le t\le 1$ $\supp_{0\le u\le1}|x(\lambda_t(u))-\bar
x_t(u)|\le \frac\delta2$. Then the relation formulated at the start of
the proof of Theorem~5 holds with $\eta=\min(\eta_1,\eta_2,\eta_3)$. But
$x(\lambda_t(u))-\bar x_t(u)=0$ if $0\le u\le t^*(t)$, and
$|x(\lambda_t(u))-\bar x_t(u)|\le \frac\delta2$ for $t^*(t)\le t\le1$
if $\eta_3>0$ is chosen so small that $t^*(t)>1-\eta_0$ for
$1-\eta_30$
$$
\liminf_{N\to\infty}(\bar\mu_N(\oo)(\bold K^\alpha)-\mu_N(\oo)(\bold
K))\ge0\quad\text{for almost all }\oo\in\Omega, \tag3.14
$$
where $\bold K^\alpha=\{x\:\rho(x,\bold K)\le \alpha\}$ is the
$\alpha$-neighborhood of the set $\bold K$.
To prove relation (3.14) we define some quantities. Let us
observe that because of Theorem~5A and Lemma~B the sequence of
probability measures $\mu'_N(\oo)$, $N=1,2,\dots$, $\mu'_N(\oo)(\bar
S_k(\cdot,\oo))=\dfrac1{\log\dfrac{B_N}{B_1}}\log\dfrac{B_{k+1}}{B_k}$,
$1\le k0$. There is some compact set $\bar {\bold
K}_0\in D([0,1])$ $\eta=\eta(\e,\alpha,\oo)$ in such a way that
$\mu'_N(\oo)(\bar{\bold K}_0)\ge 1-\frac\e2$ for all $N=1,2,\dots$.
Because of the conditions of Theorem~5 (The condition that relation
(2.14) holds) there exists some $\eta_0>0$ such that the set
$$
\bar {\bold K}_1=\{x\:x\in D([0,1]) \sup_{1-\eta_0\le
t\le1}|x(t)-x(1)|\le\frac\alpha8
$$
satisfies the inequality $\mu_0\(\bar{\bold K}_1\)\ge1-\frac\e3$. The
above defined set $\bar {\bold K}_1$ is closed, hence the compactness of
the sequence of measures $\mu'_N(\oo)$ implies that there is some
threshold $N_0=N_0(\oo)$ such that $\mu_N(\oo)\(\bar{\bold K}_1\)
\ge1-\frac\e2$ for all $N\ge N_0$. Define the set $\bar{\bold K}
=\bar{\bold K}_0\cap\bar{\bold K}_1$. Then for almost all
$\oo\in\Omega$ there is some threshold $N_0=N_0(\oo)$
$\mu_N(\oo)(\bar{\bold K})\ge1-\e$ for all $N\ge N_0$.
There exists some $\eta>0$ such that $d(x,x_t)\le\frac\alpha2$ if
$x\in\bar{\bold K}$ and $1-\eta\le t\le 1$. For all positive integers $n$
define the number $\tilde n=\tilde n(\eta)$ as
$$
\tilde n=\min\left\{k\: B_k>\(1-\frac\eta2\)B_n \text{ and } \bar
B_k>\(1-\frac\eta2\)\bar B_n\right\}. \tag3.15
$$
If the index $n$ is such that the relations $S_n(\cdot,\oo)\in\bold K$,
$d(S_n(\cdot,\oo),\bar S_n(\cdot,\oo))<\frac\alpha2$, $\bar
S_n(\cdot,\oo)\in\bar{\bold K}$ hold and $\tilde
n(n)\le m\le n$, then $\bar S_m(\cdot,\oo)\in\bold K^\alpha$. Indeed,
$$
d(\bar S_m(\cdot,\oo),\bar S_n(\cdot,\oo))<\frac\alpha2, \tag3.16
$$
since with the notation $x(\cdot)=\bar S_n(\cdot,\oo)$ we have
$\bar S_m(\cdot,\oo)=x_t(\cdot)$ with $t=\frac {B_m}{B_n}$ which
satisfies the inequality $1-\frac\eta2\le t\le1$, and this implies
(3.16). Relation (3.16) and the other conditions we have imposed imply
that $\bar S_m(\cdot,\oo))\in \bold K^\alpha$.
Let us fix some sufficiently large integer $M>0$ to be chosen later
which may depend on $\alpha$, $\e$, $\oo\in\Omega$ and the sequences
$B_n$ and $\bar B_n$, but does not depend on the index $N$ for which the
measures $\mu_N(\oo)$ and $\bar\mu_N(\oo)$ are considered. Define the
set of indices
$$
\Cal C=\Cal C(\alpha,\e,\oo,\bold K)=\left\{k\: k\ge
M,\,S_k(\cdot,\oo)\in \bold K,\,\rho(S_n(\cdot,\oo),\bar S_n(\cdot,\oo))
<\frac\alpha2,\, \bar S_n(\cdot,\oo)\in\bar{\bold K}\right\}
$$
and the sets $\bold K_0(N)\in\bold D([0,1])$
$$
\bold K_0(N)=\{S_k(\cdot,\oo)\: k\in \Cal C,\;1\le k0$ it implies relation (3.14)
hence Theorem~5.
\medskip\noindent
{\it Acknowledgement:}\/ I would like to thank Istv\'an Berkes for many
useful discussions on this subject.
\vfill\eject
\centerline{\bf Appendix}
\smallskip\noindent
{\it A simpler proof of the almost sure functional limit theorem part of
Theorem 1 by means of formula (3.2).}\smallskip\noindent
This proof is a simple adaptation of an argument of I. A. Ibragimov and
M. A. Lifshitz made in their paper {\it On almost sure type limit
theorems}.\medskip
The weak convergence of the measures $\mu_T(\oo)$ to the measure $\mu_0$
as $T\to\infty$ is equivalent to the statement
$$
\liminf_{T\to\infty}\mu_T(\oo)(\bold G)\ge \mu_0(\bold G) \quad\text
{for all open sets }\bold G.
$$
On the other hand, the following simple lemma holds:
\medskip \noindent{\bf Lemma A.} {\it Let $(M,\Cal M,\rho)$ be a
separable complete metric space with the $\sigma$-algebra $\Cal M$
generated by the topology induced by the metric $\rho$ in $M$. Let $\mu$
be a probability measure on $(M,\Cal M)$. There is a sequence of
(countably many) open subsets $\bold G_n$, $n=1,2,\dots$, of the space
$M$ in such a way that
$$
\mu(\bold G)=\sup_{\bold G_n\colon \bold G_n\subset \bold G}\mu(\bold
G_n)\quad \text{for all open sets }\bold G\subset M.
$$
}\medskip
Applying this lemma for the space $C([0,1])$ or $D([0,1])$ and the
measure $\mu_0$ the proof of the weak convergence of the probability
measures $\mu_T(\oo)$ to $\mu_0$ for almost all $\oo\in\Omega$, as
$T\to\infty$ can be reduced to the statement
$$
\liminf_{T\to\infty}\mu_T(\oo)(\bold G_n)\ge\mu_0(\bold G_n)
\quad\text{for almost all }\oo\in\Omega\text{ and } n=1,2,\dots, \tag A1
$$
where $\bold G_n$, $n=1,2,\dots$, are the open sets appearing in
Lemma~A. Indeed, this statement implies that for all open sets
$\bold G$ and $\e>0$ there exists a set $\bold G_n\subset \bold G$ such
that $\mu_0(\bold G)\le \mu_0(\bold G_n)+\e$, and
$$
\mu_0(\bold G)\le \mu_0(\bold G_n)+\e\le
\liminf_{T\to\infty}\mu_T(\oo)(\bold G_n)+\e\le
\liminf_{T\to\infty}\mu_T(\oo)(\bold G)+\e
$$
for almost all $\oo\in\Omega$. Then, by letting $\e$ tend to zero we get
the almost sure functional limit theorem.
On the other hand, defining the functionals $\Cal F_n$ in the space
$C([0,1])$ or $D([0,1])$ as $\Cal
F_n(x)=1$ if $x\in\bold G_n$, and $\Cal F_n(x)=0$ if $x\notin \bold G_n$
we get the following relation by means of formula (3.2).
$$
\lim_{T\to\infty}\mu_T(\oo)(\bold G_n)=\lim_{T\to\infty}\int\Cal
F_n(x)\mu_T(\oo)(\,dx)=\int\Cal F_n(x)\mu_0(\,dx)=\mu_0(\bold G_n)
$$
for almost all $\oo\in \Omega$, i.e.\ even a stronger version of formula
(A1) holds. This implies the weak convergence formulated in Theorem~1.
\medskip
\noindent{\it Proof of Lemma A.}
Let $x_k$, $k=1,2,\dots$, be an everywhere dense sequence in the space
$M$. Let $\bold H_{k,m}$ denote the open ball with center $x_k$ and
radius $\frac1m$ in the space $(M,\Cal M)$. Let us consider all sets
$\bold H_{k,m}$, $k=1,2,\dots$, $m=1,2,\dots$, and all finite union of
the sets $\bold H_{k,m}$. This a countable collection $\bold G_n$ of
open sets, and we claim that such a choice of the open sets $\bold
G_n$ satisfies Lemma~A.
Indeed, for all open sets $\bold G\subset M$ and $\e>0$ there exists a
compact set $\bold K\subset \bold G$ such that $\mu(\bold G)\le
\mu(\bold K)+\e$. Since all points $x\in\bold K$ have a positive
distance from the complement of the set $\bold G$, for all $x\in \bold
K$ there is a set $\bold H_{k,m}$ such that $x\in \bold
H_{k,m}\subset\bold G$. Hence the union of those sets $\bold H_{k,m}$
which are contained in $\bold G$ supply a cover of the set
$\bold K$. The compact set $\bold K$ also has a finite cover
consisting of such sets $\bold H_{k,m}$. This means that there
exists a set $\bold G_n$ such that $\bold K\subset \bold G_n\subset
\bold G$. This relation also implies that $\mu(\bold G_n)\ge\mu(\bold
K)\ge\mu\bold G)-\e$. Since such a construction can be made for all
$\e>0$ these relations imply Lemma~A.
\centerline{\bf Appendix 2.}
\medskip\noindent The argument below gives a possible measure
theoretical justification of the procedure leading to the proof of
formula (3.2).
\smallskip \noindent
We need the following results in Billingsley book~[1] (the discussion
after Theorem~8.3 for the space $C([0,1])$ and Theorem 14.5 for the
space $D([0,1])$). Put $(X,\Cal A)=\(\bold R^{[0,1]},\Cal C^{[0,1]}\)$,
where $\bold R^{[0,1]}$ is the direct product of the real line with
indices $0\le t\le1$ and $\Cal C^{[0,1]}$ is the direct product of the
usual topology on the real line with indices $0\le t\le1$. Beside this,
let us denote by $(Y,\Cal B)$ the space $C([0,1])$ or $D([0,1])$ with
the usual topology. The results quoted from Billingsley's book state
that if we denote by $\Cal M$ the $\sigma$-algebra generated by the
open sets in $(X,\Cal A)$ and by $\Cal N$ the $\sigma$-algebra
generated by the open sets in $(Y,\Cal B)$, then all $B\in \Cal N$ can
be written in the form $B=A\cap Y$ with some $A\in\Cal M$. Billingsley's
book also proves that $A\cap Y\in\Cal N$ for all $A\in\Cal M$.
Given any probability measure $\mu$ on the space $(Y,\Cal B)$ we can
define its extension $\bar\mu$ by defining $\bar\mu(C)=\mu(C\cap Y)$
for all $C\subset X$ such that $C\cap Y\in \Cal N$. The class of sets
$C$ with the property $C\subset X$ and $C\cap Y\in \Cal M$ is a
$\sigma$-algebra $\Cal G$ such that $\Cal M\subset \Cal G$, $\Cal
N\subset \Cal G$, and $\bar\mu$ is a probability measure in $\Cal G$.
Let us remark that since all $B\in \Cal N$ can be written in the form
$B=A\cap Y$ with $A\in \Cal M$, the restriction of the measure $\bar
\mu$ to $\Cal M$, determines the measure $\bar\mu$. This implies that
the finite dimensional distributions of the $C([0,1])$ or $D([0,1])$
valued stochastic process determine the distribution $\mu$ of the
process and its extension $\bar\mu$. Given a measurable function $\Cal
F$ on the space $(Y,\Cal B)$ we call its extension any measurable
function $\bar{\Cal F}$ on the space $(X,\Cal G)$ such that $\bar {\Cal
F}(y)=\Cal F(y)$ for all $y\in Y$. For instance we can define
the extension of $\Cal F$ by the formula $\bar{\Cal F}(y)=\Cal F(y)$ if
$y\in Y$, and $\bar{\Cal F}(y)=0$ if $y\notin Y$.
We can prove formula (3.2) if the functional $\Cal F$ and measures
$\mu_0$ and $\mu_T(\oo)$ are replaced by their extensions defined in
the way described above. Observe that since all trajectories
$X_t(\cdot,\oo)$ defined in (2.4) are in the space $C([0,1])$ or
$D([0,1])$ the measures $\mu_T(\oo)$ are concentrated on the set $Y$.
Then both the left and right-hand side of (3.2) remain the same if we
rewrite the original functional $\Cal F$ and measures $\mu_0$ and
$\mu_T(\oo)$ on the space $(Y,\Cal B)$ in this formula.
\bigskip\noindent {\bf References:} \medskip
\item{[1]} Billingsley, P.: Convergence of Probability measures,
Wiley \& Sons Inc.\ New York--London--Sydney--Toronto, (1968).
\item{[2]} Brosamler, G.: An almost everywhere central limit
theorem. {\it Math.\ Proc.\ Cambridge Philos.\ Soc.}\/ {\bf 104},
561--574, (1988).
\item{[3]} Dobrushin, R. L.: Gaussian and their subordinated generalized
fields {\it Annals of Probability}\/ {\bf 7} 1--28, (1979).
\item{[4]} Feller, W. An Introduction to Probability Theory and Its
Applications, Vol. II. Wiley \& Sons Inc.\ New
York--London--Sydney--Toronto, (1971).
\item {[5]} Fisher, A.: Convex invariant means and a pathwise central
limit theorem {\it Advances in Mathematics}\/ No.~3 {\bf 63} 213--248
(1987)
\item{[6]} Lacey, M. and Philipp, W.: A note on the almost everywhere
central limit theorem. {\it Statist. Prob. Letters}\/ {\bf 9}, 201--205,
(1990).
\item{[7]} Schatte, P.: On strong versions of the central limit
theorem. {\it Math. Nachr.}\/ {\bf 137}, 249--256, (1988).
\bye
~~