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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
$$
\lim_{n\to\infty}\frac1{\log
n}\sum_{k=1}^n\frac1kI\left(\frac{S_k(\oo)}{\sqrt k}<u\right)=
\Phi(u)\quad \text{for almost all } \oo\in \Omega     \tag1.1
$$
and all numbers $u$, where $I(\bold A)$ denotes the indicator function
of a set $\bold A$, and $\Phi(u)$ is the standard normal distribution
function. This result was discovered by Brosamler~[2] and Schatte~[7].
It states that appropriately normalized partial sums of iid.\ random
variables satisfy not only the central limit theorem, but for a typical
$\oo\in \Omega$ the weighted averages of the functions $g_k(u,\oo)=
I\(S_k(\oo)<u\sqrt k\)$ with appropriate weights converge to the
normal law. Later this result was formulated in a more general form
which states that not only the weighted averages of the functions
$I\(S_k(\oo)<u\sqrt k\)$ converge to the normal distribution
function for a typical $\oo$, but a similar result also holds for
sequences of random broken lines or polygons $G_n(u)=G_n(u,\oo)$,
$n=1,2,\dots$, defined in an appropriate way on the interval $[0,1]$
by means of the partial sums $S_1(\oo),\dots,S_n(\oo)$.
 
Define a random measure $\mu_n=\mu_n(\oo)$ for all $n$ by attaching an
appropriate weight $a_k=a_{k,n}$ to the functions $G_k(u,\oo)$ for all
$1\le k\le n$. Then these measures converge weakly to the Wiener
measure for almost all $\oo$. Such a result is called an almost sure
functional limit theorem. Later we formulate this notion in a more
detailed form.
 
The almost sure central (and also the functional) limit theorem shows
some similarity to the ergodic theorem which states --- in physical
terminology --- that the space and time averages of ergodic sequences
agree. In the case of the almost sure central limit theorem an analogous
result holds for the normalized partial sums $\dfrac{S_k(\oo)}{\sqrt
k}$, $k=1,2,\dots$. Now the time average is replaced by a weighted
time average, where the $k$-th term gets weight $a_k=a_{k,n}=\dfrac1
{\log (n+1)}\log \dfrac{k+1}k\sim\dfrac 1{k\log n}$, $1\le k\le n$, in
the $n$-th block instead of the weight $\dfrac1n$ given to the first
$n$ terms in the ergodic theorem. On the other hand, $\dfrac{S_n(\oo)}
{\sqrt n}$ is asymptotically normally distributed, with expectation
zero and variance one. Hence the right-hand side in formula (1.1)
equals $\limm_{n\to\infty}E I\(\dfrac{S_n(\oo)}{\sqrt n}<u\)$, and
this expression resembles to a space average. This similarity of the
almost sure central limit theorem to the ergodic theorem may be put even
stronger by an appropriate time scaling to be explained later.
 
The relation between the ergodic theorem and almost sure central (and
functional) limit theorem is deeper than the above mentioned formal
analogy. It was pointed out, --- by our knowledge it was discovered by
Brosamler~[1], Fisher~[5] and Lacey and Philipps in [6] --- that
these theorems can be deduced from the ergodic theorem applied to the
Ornstein--Uhlenbeck process.
 
In the present paper we discuss how the almost sure central and
functional limit theorem can be generalized and proved by means of the
ergodic theorem in a natural way. The proof has two main ingredients.
The first one is to show that a result analogous to the almost sure
functional limit theorem holds for the Wiener process. This can be
deduced from the ergodic theorem for the Ornstein--Uhlenbeck process.
This is an ergodic process which can be obtained from the Wiener
process by means of a well-known transformation. The second ingredient
is to show that, since the random polygons or broken lines constructed
from the partial sums of independent random variables in a natural way
behave similarly to the Wiener process, the almost sure central limit
theorem for the Wiener process also implies this result for the random
polygons (or broken lines) made from normalized partial sums of
independent random variables.
 
First we show that the method of proving the almost sure functional
limit theorem for the Wiener process by means of the ergodic theorem for
the Ornstein--Uhlenbeck process can be generalized for a large class of
other processes, for the so-called self-similar processes. The
stationarity property of the Ornstein--Uhlenbeck process is equivalent
to the self-similarity property of the Wiener process, a property which
holds for all self-similar processes. Actually, self-similar
process are those processes which appear as the limit in different limit
theorems. Similarly to the construction of the Ornstein--Uhlenbeck
process generalized Ornstein--Uhlenbeck
processes can be constructed as the transforms of self-similar
processes. These generalized Ornstein-Uhlenbeck processes are
stationary processes, and the application of the ergodic theorem for
them enables us to prove the almost sure functional limit theorem for
general self-similar processes. Then with the help of some further
work we can also prove the almost sure functional limit theorem for
their appropriate discretized versions.
 
In the next step we want to find a good coupling argument which enables
us to prove the almost sure invariance principle not only for
(self-similar) limit processes but also for processes in the domain of
their attraction. To carry out such a program a coupling argument has
to be introduced which is adapted to the present problem. We shall do
it by introducing a notion we call the Property~A.
 
In Part II.\ of this work we shall prove the almost sure functional
limit theorem for independent random variables whose partial
sums converge to the normal or to a stable law. In the proofs we shall
exploit that the Wiener process and the stable process are
self-similar, hence the results of the present paper can be applied for
them. Then we can prove, by applying the coupling argument of the
present paper, the almost sure invariance principle for independent
random variables which satisfy certain (weak) conditions.
 
There are other processes which are natural candidates for almost sure
functional limit theorem type results, e.g.\ random processes in the
domain of attraction of a self-similar process subordinated to a
Gaussian process (see Dobrushin~[3]). But such problems will not be
discussed here.
 
Several results of the present paper can be traced down in earlier
works. Our main goal is to explain the main ideas behind these results
and to present a unified treatment of various problems in this subject.
The first part of this work considers general results where no
independence type condition is assumed. In the second part different
arguments --- the techniques worked out for the study of independent
random variables --- are applied, and we deal there with almost sure
functional limit theorems for independent random variables. This paper
consists of three sections. In Section~2 we formulate the main results,
and Section~3 contains the proofs.
\medskip\noindent
{\bf  2. The main results of the paper}
\medskip\noindent
To formulate our results first we recall the definition of self-similar
processes with self-similarity parameter $\alpha$ and define with their
help a new process which we call a generalized Ornstein--Uhlenbeck
process.
\medskip\noindent
{\bf Definition of self-similar processes.} {\it We call a stochastic
process $X(u,\oo)$, $u\ge 0$,  $X(0,\oo)\equiv0$, self-similar with
self-similarity parameter $\alpha$, $\alpha>0$, 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 $Z(t,\oo)$,
$-\infty<t<\infty$, defined by formula
$$
Z(t,\oo)=\frac{X(e^t,\oo)}{e^{t/\alpha}},\quad -\infty<t<\infty,
\tag2.2
$$
the generalized Ornstein--Uhlenbeck process corresponding to the process
$X(u,\oo)$.}\medskip
Let us remark that the generalized Ornstein--Uhlenbeck process
corresponding to the Wiener process is the usual Ornstein--Uhlenbeck
process.
 
A Wiener process $W(t,\oo)$, $t\ge 0$, has continuous trajectories, the
trajectories of a stable process $X(t,\oo)$ are so-called c\`adl\`ag
(continue \`a droite, limite \`a gauche) functions, i.e. all
trajectories $X(\cdot,\oo)$ are continuous from the right, and have a
left-hand side limit in all points $t>0$. 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 $Z(t,\oo)$,
$-\infty<t<\infty$, is stationary. Let us assume that the process
$Z(t,\oo)$ is not only stationary, but also ergodic.
Then for all measurable and bounded functionals $\Cal F$ on the
space $C([0,1])$ or $D([0,1])$ (depending on whether the trajectories
of $X(\cdot,\oo)$ are continuous or only c\`adl\`ag functions)
$$
\lim_{T\to\infty}\frac1{\log T}\intl_1^T \frac1t \Cal F(X_t(u,\oo))\,dt=
E\Cal F(X_1(u,\oo))\quad \text{for almost all }\oo, \tag2.3
$$
where
$$
X_t(u,\oo)=\dfrac{X(ut,\oo)}{t^{1/\alpha}}, \quad 0\le u\le1,\;t>0.
\tag2.4
$$
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<t\le1,
\tag$2.4'$
$$
where $\alpha$ is the self-similarity parameter of the underlying
self-similar process $X(\cdot,\oo)$. Then for almost all
$\oo\in \Omega$
$$
\lim_{\e\to0}\lim_{T\to\infty}\mu_T(\oo)\(\sup_{1-\e\le s,t\le 1}
\rho(x_s,x_t)>\delta\)=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
$1=B_{N,1}<B_{N,2}<\cdots<B_{N,k_N}=B_N$ of the interval $[1,B_N]$. Let
us define the random variable $\hat \xi(t)$ such that
$\hat\xi(t)=\xi(B_{k,N})=X_{B_{k,N}}(\cdot,\oo)$ if
$t\in[B_{k,N},B_{k+1,N}]$. This random variable is close to the
previously defined random variable $\xi(t)$, hence it is natural to
expect that if $\hat\mu_{B_n}(\oo)$ denotes its distribution, then
the measures $\hat\mu_{B_N}(\oo)$ have the same weak limit as the
measures $\mu_{B_N}(\oo)$ as $N\to\infty$. The first statement
of Theorem~3 is a result of this type. Then we prove that an
appropriate small modification of the functions
$\xi(B_{k,N})=X_{B_k,N}(\cdot,\oo)$ does not change the limit behaviour
of the measures $\hat\mu_{B_N}(\oo)$. The second statement of Theorem~3
is such a result.
\medskip\noindent
{\bf Theorem 3.} {\it Let us assume that the conditions of
Theorem~1 and Theorem~2 are satisfied. For all $N=0,1,\dots$ let us
consider a finite increasing sequence of real numbers
$1=B_{1,N}<B_{2,N}<\cdots <B_{k_N,N}$, and for the sake of simpler
notation let us denote $B_{k_N,N}$ by $B_N$. Let us assume that these
sequences satisfy the following properties:
$$
\aligned
\limm_{N\to\infty}B_N=\infty, \quad
&\limm_{N\to\infty}\dfrac {\log B_{j,N}}{\log B_N}=0 \text{ for
all fixed }j,\\
&\qquad\text{and}\quad \limm_{j\to\infty}\supp_{(k,N)\:j\le k<N
}\dfrac{B_{k+1,N}}{B_{k,N}}=1.
\endaligned \tag2.7
$$
Moreover, assume the following strengthened form
of the relation $\limm_{N\to\infty}B_N=\infty$:
$$
\lim_{j\to\infty}\,\inf_{N\:N\ge j}\frac{B_{l,n}}{B_{j,N}}=\infty
\quad\text{for all fixed } l=1,2,\dots. \tag2.8
$$
For all $\oo\in \Omega$ define the (random) measures $\hat\mu_N(\oo)$,
$N=1,2,\dots$, with the help of the sequences
$1=B_{1,N}<B_{2,N}<\cdots<B_{k_N,N}$ in the following way:
 
The measure
$\hat\mu_N(\oo)$, $N=1,2,\dots$, is concentrated on the trajectories
$X_{B_{j,N}}(\cdot,\oo)$, $1\le j< k_N$, where
$X_t(\cdot,\oo)$ is defined in (2.4), and
$$\gathered
\hat\mu_N(\oo)(X_{B_{j,N}}(\cdot,\oo))=\frac1{\log
B_N}\intl_{B_{j,N}}^{B_{j+1,N}}\frac 1u\,du=\frac1{\log
B_N}\log\frac{B_{j+1,N}}{B_{j,N}},\\
1\le j< k_N. \endgathered
\tag2.9
$$
Then for almost all $\oo$ the measures $\hat\mu_N(\oo)$ converge weakly
to $\mu_0$.
 
For all $\oo\in\Omega$ let us also define the following random broken
lines
$\bar X_{B_{j,N}}(\cdot,\oo)$ which are ``discretizations"
of the trajectories $X_{B_{j,N}}(\cdot,\oo)$.
$$
\align
\bar X_{B_{j,N}}(s,\oo)&=X_{B_{j,N}}\(\frac{B_{l-1,N}}{B_{j,N}},\oo\)
\quad
\text{if}\quad \frac{B_{l-1,N}}{B_{j,N}}\le s<\frac{B_{l,N}}{B_{j,N}},\\
&\qquad 1\le l\le j,\;\;1\le j< k_N,
\quad\text{and }\bar X_{B_{j,N}}(1,\oo)=X_{B_{j,N}}(1,\oo),
\endalign
$$
where $B_{0,N}=0$. (The definition $B_{0,N}=0$ is needed to
 define $\bar X_{B_{j,N}}(s,\oo)$ also for $0\le sB_{j,N}<B_{1,N}$.)
 
Define the measures $\bar\mu_N(\oo)$  (with the help of the already
defined measures $\hat \mu_N(\oo)$) as
$$
\bar\mu_N(\oo)(\bar X_{B_{j,N}}(\cdot,\oo))=
\hat\mu_N(\oo)(X_{B_{j,N}}(\cdot,\oo))=\frac1{\log B_N}
\log\frac{B_{j+1,N}}{B_{j,N}},\quad 1\le j< k_N. \tag$2.9'$
$$
Then for almost all $\oo\in\Omega$ the probability measures
$\bar\mu_N(\oo)$ converge weakly to the probability  measure $\mu_0$
as $N\to\infty$.}
\medskip
We have defined $\bar X_{B_{j,N}}(\cdot,\oo)$ as a broken line
with discontinuities and not as a polygon where the values of
$X_{B_{j,N}}$ in the points $\dfrac{B_{l,N}}{B_{j,N}}$ are connected
by linear segments. The reason for working with broken lines is that
we want to prove results which are valid also in the case when the
processes $X_t(\cdot,\oo)$ take their values in $D([0,1])$ but not
necessarily in the space $C([0,1])$. In the general case the results
we want to prove are valid only when broken lines are considered. In
the case of processes with continuous trajectories we also could have
defined them as random polygons. Moreover, it follows from some results
of the general theory (see e.g.\ Section~18 in Billingsley's book~[1])
that if the distribution of the processes consisting of the above
defined random broken lines converge to a measure in the $C([0,1])$
space, then the distributions of the naturally defined random polygon
version of these processes have the same limit in the $C([0,1])$ space.
 
Let $\xi_n(\oo)$, $n=1,2,\dots$, be a sequence of random variables,
and let us define the partial sums $S_n(\oo)=\summ_{k=1}^n\xi_k(\oo)$,
$n=1,2,\dots$, $S_0(\oo)\equiv0$. Let us also consider two appropriate
monotone increasing numerical sequences $A_n$ and $B_n$, $n=0,1,\dots$,
of positive numbers such that
$$
B_0=0,\quad \limm_{n\to\infty}A_n=\infty,\quad
\limm_{n\to\infty}B_n=\infty,\quad  \text{and}\quad
\limm_{n\to\infty}\dfrac{B_{n+1}}{B_n}=1. \tag2.10
$$
For all $k=1,2,\dots$ let us consider the partition $0=s_{0,k}\le
s_{1,k}\le\cdots\le s_{k,k}$ of the interval $[0,1]$, defined by the
formula $s_{j,k}=\dfrac{B_j}{B_k}$, $0\le j\le k$. Let us also define
with the help of the quantities $\xi_n(\oo)$, $A_n$ and $B_n$,
$n=1,2,\dots$ the following random broken lines $S_k(s,\oo)$, $0\le
s\le 1$, $k=1,2,\dots$,
$$
S_k(s,\oo)=\frac{S_{j-1}(\oo)}{A_k}
\quad\text{if } s_{j-1,k}\le s< s_{j,k}, \;1\le j\le k,\quad
S_k(1,\oo)=\frac {S_k(\oo)}{A_k} \tag2.11
$$
Now we introduce the following definition.
\medskip\noindent
{\bf Definition of the almost sure functional limit theorem.}
{\it Let $\xi_n(\oo)$, $n=1,2,\dots$, be a sequence of random
variables, and let two monotone increasing sequences of non-negative
real numbers $A_n$ and $B_n$, $n=1,2,\dots$, be given
which satisfy formula (2.10). Let us consider the random broken lines
$S_k(s,\oo)$, $0\le s\le 1$, defined with the help of their partial sums
$S_k(\oo)$, $k=1,2,\dots$, by formula (2.11). For all $\oo\in\Omega$ and
$N=1,2,\dots$, define the random measure $\mu_N(\oo)$ in the following
way: The measure $\mu_N(\oo)$ is concentrated on the random broken
lines $S_k(\cdot,\oo)$, $1\le k<N$, and
$$
\mu_N(\oo)(S_k(\cdot,\oo))
=\frac1{\log\dfrac{B_N}{B_1}}\log\frac{B_{k+1}}{B_k},\quad 1\le k<
N. \tag2.12
$$
We say that the sequence of random variables $\xi_n(\oo)$,
$n=1,2,\dots$, satisfies the almost sure functional limit theorem with
weight functions $A_n$ and $B_n$, $n=1,2,\dots$, and limit measure
$\mu_0$ on the space $D([0,1])$ if for almost all $\oo\in\Omega$ the
probability measures $\mu_N(\oo)$ converge weakly to the measure
$\mu_0$ as $N\to\infty$. In the special case when the limit measure
$\mu_0$ is the Wiener measure we say that these random variables
satisfy the almost sure functional central limit theorem.}
\medskip
If the limit measure $\mu_0$ is concentrated in the space $C([0,1])$,
then the broken lines $S_k(\cdot,\oo)$ can be replaced by a natural
modification which is a random polygon. Then we can consider a version
of the measures $\mu_N(\oo)$ which are defined in the same way as the
original ones, only the random processes $S_k(\cdot,\oo)$ are replaced
by their random polygon version. Then the convergence of the original
measures $\mu_N(\oo)$ to $\mu_0$ in the space $D([0,1])$ implies the
convergence of their modified version in the $C([0,1])$ space with
the same limit. Let us also remark that although we allowed fairly
large freedom in the definition of the sequence $A_n$ in the definition
of the almost sure functional limit theorem, nevertheless we shall
always choose it in a very special way. Namely, in all almost sure
functional limit theorems we shall prove the limit measure is the
distribution of a self-similar process with a self-similarity parameter
$\alpha>0$ 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 $-\infty<T<\infty$. Hence the process $Z(t,\oo)$,
$-\infty<t<\infty$, is stationary. If it is not only stationary,
but also ergodic, then the ergodic theorem can be applied for the
process $Z(\cdot,\oo)$ and all bounded and measurable functionals $\Cal
G$ on the space $(R^{(-\infty,\infty)},\Cal B,\mu)$, where
$R^{(-\infty,\infty)}$ is the space of functions on the interval
$(-\infty,\infty)$, $\Cal B_0$ is the $\sigma$-algebra induced by the
usual Borel (product) topology on $R^{(-\infty,\infty)}$, $\mu$ is the
distribution of the process $Z(\cdot,\oo)$ on the space
$(R^{(-\infty,\infty)},\Cal B_0)$, and $\Cal B$ is the closure of the
$\sigma$-algebra $\Cal B_0$ with respect to the measure $\mu$. This
means that $\bold B\in \Cal B$ if and only if there exists some
$\bold B_0\in\Cal B_0$ such that $\mu(\bold B_0\Delta \bold B)=0$
for the symmetric difference $\bold B_0\Delta \bold B$, or more
precisely there is a $\Cal B_0$ measurable set $\bold C$ such that
$\mu(\bold C)=0$ and $\bold B_0\Delta \bold B\subset\bold C$.
Furthermore, we introduce the shift operators $\bold T_s$ defined by
the formula $\bold T_s(z(\cdot))=z(s+\cdot)$ for all $z(\cdot)\in
R^{(-\infty,\infty)}$ and put $Z_s(v,\oo)=Z(s+v,\oo)$,
$-\infty<v<\infty$. Then the ergodic theorem implies that
$$
\aligned
\lim_{T\to\infty}\frac1{\log T}\intl_{0}^{\log T}\Cal G(\bold T_s
(Z(u,\oo))\,ds&=\lim_{T\to\infty}\frac1{\log T}\intl_{0}^{\log T}
\Cal G(Z_s(u,\oo))\,ds\\
&=E\Cal G(Z(u,\oo))\quad\text{for almost all
}\oo\in\Omega.
\endaligned \tag3.1
$$
 
Given a bounded measurable functional
\plainfootnote{*}{\small Yurij Davydov suggested to show that the
procedure we follow is legitime also from measure theoretical point of
view, i.e.\ no problem arises because $\Cal F$ is defined in the space
$C([0,1])$ or $D([0,1])$, and we work in the space of all functions in
the interval [0,1] with the usual product topology and
$\sigma$-algebra. This procedure can be justified for instance by
means of the results proved in Billingsley's book~[1], in the discussion
after Theorem~8.3 and in Theorem 14.5. These results state that the
$\sigma$-algebra defined by the usual topology in the spaces $C([0,1])$
and $D([0,1])$ agrees with the restriction of the usual $\sigma$-algebra
in the space of all functions on the interval [0,1] to these spaces.
We give a more detailed explanation in Appendix~2.}
$\Cal F$ on the space $C([0,1])$ or $D([0,1])$ let us extend it to the
space of all measurable functions on the space $R^{[0,1]}$ of all
functions on the interval $[0,1]$ by defining $\Cal F(x)=0$ if the
function $x=x(\cdot)$ is not in the space $C([0,1])$ or $D([0,1])$.
Then we define the functional $\Cal G=\Cal G(\Cal F)$ on the space
$R^{(-\infty,\infty)}$ by the formula $\Cal G(z)=\Cal F(x_z)$ with
$x_z(u)=u^{1/\alpha}z(\log u)$, $0<u\le1$, $z(0)=0$. We can write
$$
\frac1{\log T}\intl_1^T \frac1t \Cal F(X_t(\cdot,\oo))\,dt=
\frac1{\log T}\intl_0^{\log T}\Cal F(X_{e^s}(\cdot,\oo))\,ds
=\frac1{\log T}\intl_0^{\log T}\Cal G(Z_s(\cdot,\oo))\,ds,
$$
since $\Cal G(Z_s(\cdot,\oo))=\Cal F(X_{e^s}(\cdot,\oo))$. Indeed,
$$
\align
x_{Z_s(\cdot,\oo)}(u)&=u^{1/\alpha}Z_s(\log u,\oo)=u^{1/\alpha}Z(s+\log
u,\oo)=u^{1/\alpha} \frac{X(e^{s+\log u},\oo)}{e^{(s+\log u)/\alpha}}\\
&=\frac{X(ue^s,\oo)}{e^{s/\alpha}},\quad\text {for all } 0\le u\le1,
\endalign
$$
hence $x_{Z_s(\cdot,\oo)}=X_{e^s}(\cdot,\oo)$, where $X_s(\cdot,\oo)$
was defined in~(2.4). This relation (with the choice $s=0$) implies in
particular that
$$
E\Cal G(Z(\cdot,\oo))=E\Cal G(Z_0(\cdot,\oo))=E\Cal F(X_1(\cdot,\oo)).
$$
These identities together with relation (3.1) and
the definition of the measures~$\mu_T(\oo)$ introduced in the
formulation of Theorem~1 imply that
$$
\aligned
\lim_{T\to\infty}\intl \Cal F(x)\,d\mu_T(\oo)(x)
&=\lim_{T\to\infty}\frac1{\log T}\intl_1^T \frac1t \Cal
F(X_t(\cdot,\oo))\,dt\\
&=\lim_{T\to\infty}\frac1{\log T}\intl_0^{\log T}
\Cal G(Z_s(\cdot,\oo))\,ds=E\Cal G(Z(\cdot,\oo))   \\
&=E\Cal F(X_1(\cdot,\oo))=\intl \Cal F(x)\,d\mu_0(x)
\quad\text{for almost all }\oo\in\Omega.
\endaligned \tag3.2
$$
To prove Theorem~1 we have to show that relation (3.2) holds
simultaneously for all bounded and continuous functionals $\Cal F$ for
almost all $\oo\in\Omega$, and the exceptional set of $\oo\in\Omega$ of
measure zero should not depend on the functional $\Cal F$. We prove this
\plainfootnote{*}{{\small After having published this
work I learned about the still
unpublished paper ``On almost sure type limit theorems" (in Russian) of
I.~A.~Ibragimov and M.~A.~Lifshitz, where a much
simpler proof of this statement is presented. I added an Appendix to
this paper which supplies their proof. This can replace the remaining
part of the proof of Theorem~1 except the last part, where
the ergodicity of certain processes is proved.}}
with the help of the following \medskip\noindent
{\bf Lemma A.} {\it Under the conditions of Theorem 1 the closure
of the set of (random) measures $\mu_T(\oo)$, $T\ge1$, are compact in
the topology defining weak convergence of probability measures in the
space $C([0,1])$ or $D([0,1])$ (depending on where the distribution of
the process $X(\cdot,\oo)$ is defined) for almost all $\oo\in
\Omega$.}
\medskip\noindent
{\it Proof of Lemma A:}\/  We apply the result that a set of
probability measures $\mu_T$ on a separable complete metric space
(endowed with the topology inducing weak convergence) is compact if
and only if for all
$\e>0$ 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 $0<a\le1$,
$$
\mu_t(\oo)\(x\:\supp_{0\le u\le1}|x(u)|\ge K\)\le
\mu_{T_0(\oo)}(\oo)\(x\:\supp_{0\le u\le1}|x(u)|\ge K
T_0(\oo)^{-1/\alpha}\),
$$
and
$$
\align
&\mu_t(\oo)\(x\in C([0,1])\:|w_x(\delta)|\ge \eta\) \\
&\qquad \le\mu_{T_0(\oo)}(\oo)\(x\in C([0,1])\:
|w_x(\delta T_0(\oo))|\ge \eta T_0(\oo)^{-1/\alpha}\).
\endalign
$$
if $1\le t\le T_0(\oo)$. These probabilities can be taken small by
choosing a sufficiently large $K>0$ 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
$w_x[a,b)=\supp_{a\le s,t<b}|x(t)-x(s)|$ for all numbers $0\le a<b\le1$.
 
The proof of formula $(3.3')$ is similar to that of formula~(3.3). Let
us introduce the functionals
$$
\align
&\Cal F_1(x)=I\(\supp_{0\le
t\le1}|x(t)|\ge K\),\quad \Cal F_2(x)=I\(w''_x(\delta)\ge\eta\),\\
&\Cal F_3(x)=I\(w_x[0,\delta)\ge\eta\)\quad\text{and }
\Cal F_4(x)=I\(w_x[1-\delta,1)\ge\eta\)
\endalign
$$
on the space $D([0,1])$, where
the constants $K=K(\e)$ and $\delta=\delta(\e,\eta)$ will be
appropriately chosen. Let us observe that with their appropriate choice
we can achieve that $E\Cal F_i(X(\cdot,\oo)\le \e^2$ for $i=1,2,3,4$.
To see this it is enough to observe that for all $x\in D([0,1])$
$\supp_{0\le t\le1}|x(t)|<\infty$, $\limm_{\delta\to0}w''_x(\delta)=0$
(see e.g.\ formulas (14.8) and (14.46) in Billingsley's book~[1]),
$\limm_{\delta\to0}w_x[0,\delta)=0$ and
$\limm_{\delta\to0}w_x[1-\delta,1)=0$. These functionals $\Cal F_i$
take values 0 and 1, and formula $(3,3')$ can be proved similarly to
(3.3) with the help of relation~(3.2). In such a way Lemma~A is proved.
\medskip
Now we turn back to the proof of Theorem~1. We prove with the help of
Lemma~A, formula (3.2) and  a compactness argument that for almost all
$\oo\in\Omega$ the sequence of measures $\mu_T(\oo)$ converges weakly
to $\mu_0$ as $T\to\infty$. First we show that for all $\e_0>0$ 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 $-\infty<T<\infty$ have probability zero or
one. This is a property which is actually stronger than the ergodicity
of the process.
\medskip\noindent
{\it Proof of Theorem 2:}\/ Theorem~2 will be
proved by means of formula (3.2) with an appropriately defined
functional $\Cal F$ in the space $C([0,1])$ or $D([0,1])$. Let us
define the functional $\Cal F=\Cal F_{\e,\delta}$ with some $\e>0$ 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 s<B_{j+1,N}, \quad \text{ and }
X_{B_{j,N}}(\cdot,\oo)\in \bold B\},
\endalign
$$
where the measure $\bar\lambda_T$ was defined in the formulation of
Lemma~1.
 
For a pair of numbers $\e>0$ and $\eta>0$ define the set
$$
\bold A(\e,\eta)=\left\{x\in D([0,1])\: \supp_{1-\eta<s\le t\le 1}
\rho(x_s,x_t) \le\e\right\}.
$$
Given some  $\e>0$ 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
$j_0\le k<N$, and  $\dfrac{\log B_{j_0,N}}{\log B_N}<\delta$ if $N\ge
N_1$. Then for all $N\ge N_1$
$$
\align
&\hat\mu_N(\oo)(\bold F^\e) \ge
\hat\mu_N(\oo)(X_{B_{k,N}}(\cdot,\oo)\in \bold F^\e,\text{ for some } k
\ge j_0)\\
&\qquad=
\bar\lambda_{B_N}(\{s\: \text{there is some } j_0\le j< k_N\text
{ such that } \\
&\hskip 5truecm B_{j,N}\le s<B_{j+1,N}\text { and }
X_{B_j,N}(\cdot,\oo)\in \bold F^\e\})\\
&\qquad \ge \bar\lambda_{B_N}(\{s\:B_{j_0,N}\le s<B_{N}\text { and }
X_{s}(\cdot,\oo)\in \bold F\cap \bold A(\e,\eta)\})
\endalign
$$
The last inequality in this relation holds, because, in the case
when $X_s(\cdot,\oo)\in \bold F\cap \bold A_N$ and $s\in [B_{j,N},
B_{j+1,N})$ with some $j_0\le j<k_N$ (observe that the relation
$[B_{j_0,N},B_N)=\bigcupp_{j=j_0}^{k_N-1}[B_{j,N},B_{j+1,N})$ holds),
then $X_{B_{j,N}}(\cdot,\oo)\in \bold F^\e$, and this
implies that all points $s\in (B_{j,N},B_{j+1,N}]$ are contained
in the set whose $\bar\lambda_T$ measure is considered in the previous
expression. To see the validity of this statement observe that with the
notation $x=X_s(\cdot,\oo)$, $x\in D([0,1])$
$X_{B_{j,N}}(\cdot,\oo)=x_u$ with $u=\dfrac{B_{j,N}}s$, which satisfies
the inequality $1-\eta\le \dfrac1{1+\frac\eta 2}\le u\le1$, where the
function $x_u$ is defined in formula $(2.4')$. Hence $x\in \bold
A(\e,\eta)\cap\bold F$ implies that $x_u\in \bold F^\e$, as we claimed.
Then we get that
$$
\aligned
\hat\mu_N(\oo)(\bold F^\e) &\ge\bar \lambda_{B_N}(s\: s\in [1,B_{N}),
\text { and }X_s(\cdot,\oo)\in \bold F)\\
&\qquad-\bar\lambda_{B_N}([1,B_{j_0,N}))-\mu_{B_N}(D([0,1])\setminus
\bold A(\e,\eta))\\
&\ge  \bar \lambda_{B_N}(s\: s\in [1,B_{N}),
\text { and }X_s(\cdot,\oo)\in \bold F)-2\delta=\mu_{B_N}(\bold
F)-2\delta, \endaligned \tag3.9
$$
because $\mu_{B_N}(D([0,1])\setminus \bold A(\e,\eta))\le \delta$ and
$$
\bar \lambda_{B_N}([1,B_{j_0,N}))=\dfrac1{\log B_N}
\intl_1^{B_{j_0,N}}\dfrac1t\,dt=\dfrac{\log
B_{j_0,N}}{\log B_N}\le\delta.
$$
Letting $\delta\to0$ in formula (3.9) we get
formula (3.8). This implies the first part of Theorem~3.
 
We prove the second statement of Theorem~3 with the help of the
Corollary of Lemma~B, where $\hat\mu_N(\oo)$ plays the role of $\mu_N$
and $\bar\mu_N(\oo)$ the role of $\bar\mu_N$. We define the measure
$P_N^\e=P_N(\oo)$ on the space $D([0,1])\times D([0,1])$ independently
of the parameter $\e$ in the following way: The measure $P_N(\oo)$ is
concentrated on the trajectories $(X_{B_j,N}(\cdot,\oo), \bar
X_{B_j,N}(\cdot,\oo))$, and
$$
P_N(\oo)((X_{B_j,N}(\cdot,\oo),\bar
X_{B_j,N}(\cdot,\oo))=\dfrac1{\log B_N}\log\dfrac{B_{j+1,N}}{B_{j,N}}.
$$
Such a coupling can be constructed e.g.\ in the following way: For all
$N=1,2,\dots$ let $A_N$ denote the set $\bold A_N=\{1,\dots,k_N\}$,
$\Cal A_N$ the $\sigma$-algebra consisting of all subsets of $\bold
A_N$, and define the probability measure $\nu_N$, $\nu_N(j)=\dfrac1
{\log B_N}\log\dfrac{B_{j+1,N}}{B_{j,N}}$, $1\le j<k_N$ on $(\bold
A_N,\Cal A_N)$. Then for all $\oo\in\Omega$ define the random variable
$\xi_\oo(j)=(X_{B_j,N}(\cdot,\oo),\bar X_{B_j,N}(\cdot,\oo))$, $1\le
j\le k_N$, on the probability space $(\bold A_N,\Cal A_N,\nu_N)$, and
let $P_N(\oo)$ be the distribution of the random variables $\xi_\oo$ in
the space $D([0,1])\times D([0,1])$.
 
The marginal distributions of the measures $P_N(\oo)$ are
$\hat\mu_N(\oo)$ and $\bar\mu_N(\oo)$. Hence by Corollary of Lemma~B
it is enough to prove that for almost all $\oo$ the relation
$$
\limm_{N\to\infty}P_N(\oo)(\bold A_N(\e,\oo))=0 \tag3.10
$$
holds with
$$
\bold
A_N(\e,\oo)=\left\{(X_{B_j,N}(\cdot,\oo),\bar X_{B_j,N}(\cdot,\oo))\:
\rho(X_{B_j,N}(\cdot,\oo),\bar X_{B_j,N}(\cdot,\oo))>\e\right\}
$$
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 $0=t_{j,0,N}<t_{j,1,N}<\cdots<t_{j,j,N}=1$ of the
interval $[0,1]$ such that $\bar X_{B_{j,N}}(t,\oo)=
X_{B_{j,N}}(t_{j,l-1,N},\oo)$ if $t_{j,l-1,N}\le t<t_{j,l,N}$, $1\le
l\le j$, and $\bar X_{B_{j,N}}(1,\oo)=X_{B_{j,N}}(1,\oo)$. The numbers
$t_{j,l,N}$ could be given explicitly as $t_{j,l,N}=\dfrac{B_{l-1,N}}
{B_{j,N}}$, but we do not need their explicit form. What we need is
the fact that conditions (2.7) and (2.8) imposed on the numbers
$B_{j,N}$ imply
that $\limm_{\hat\jmath\to\infty}\,\limsupp_{N\to\infty}\, \supp_{N\ge
j\ge\hat \jmath}\,\supp_{1\le l\le j}(t_{j,l,N}-t_{j,l-1,N})=0$. This
relation holds since for all $\eta>0$ 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 $\delta>0$:
$$
g(x,\delta)=\sup\Sb 0=t_0<t_1<\cdots<t_s=1\\ t_j-t_{j-1}\le
\delta,\; j=1,\dots,s \endSb \rho(x,\bar x_{t_0,\cdots,t_s}), \tag3.12
$$
where
$$
\bar x_{t_0,\cdots,t_s}(t)=x(t_{j-1}) \text{ if } t_{j-1}\le t<t_j,\;
j=1,\dots,s \quad\text{and}\quad \bar x_{t_0,\cdots,t_s}(1)=x(1).
$$
We shall prove the following Lemma~C which is probably well-known among
experts, but whose explicit formulation we did not find in the
literature.
\medskip\noindent
{\bf Lemma C.} {\it For all functions $x\in D([0,1])$
$\limm_{\delta\to 0}g(x,\delta)=0$. Moreover, for all compact sets
$\bold K\subset D([0,1])$
$$
\lim_{\delta\to 0}\sup_{x\in\bold K} g(x,\delta)=0.
$$
}\medskip
Then to finish the proof of Theorem~3 it is enough to show that
$\limm_{\delta\to0}\supp_{x\in \bold K}g(x,\delta)=0$
for all compact
sets $\bold K\subset D([0,1])$, where the function $g(x,\delta)$ is
defined in formula (3.12), and this is the content of Lemma~C.
\medskip\noindent
{\it Proof of Lemma  C.}\/  It is known (see e.g. Billingsley's book
[1] formulas (14.6) and (14.7)) that for all $\eta>0$ there is some
$\alpha=\alpha(\eta)>0$ and a partition $0=u_0<u_1<\cdots<u_r=1$ of
the interval $[0,1]$ such that for $u_j-u_{j-1}>\alpha$, and
$\supp_{1\le j\le r}\supp_{u_{j-1}\le s,t<u_j}|x(s)-x(t)|<\eta$.
Let us consider an arbitrary partition $0=t_0<t_1<\cdots<t_s=1$ of the
interval $[0,1]$ such that $\supp_{1\le j\le s}|t_j-t_{j-1}|
<\alpha\eta$. We claim that in this case $\rho(x,\bar
x_{t_1,\dots,t_s})\le \eta$. Since this relation holds for
all $\eta>0$, it implies the first statement of Lemma~C.
 
To prove this statement let us consider the partition
$0=T_0<T_1<\cdots<T_r$, such that the interval $[T_j,T_{j+1})$ is the
union of those intervals $[t_l,t_{l+1})$ for which $t_l\in
[u_j,u_{j+1})$. Let $\lambda(\cdot)$ be that mapping of the
interval $[0,1]$ into itself which maps the interval $[u_j,u_{j+1})$
linearly to the interval $[T_j, T_{j+1})$. Then $\supp_{0\le u\le
1}|x(\lambda(u))-\bar x_{t_1,\dots,t_s}(u)|\le \eta$, and also
$\supp_{t\neq s}\log\left|\dfrac{\lambda(t)-\lambda(s)}{t-s}\right|
\le\eta$. Hence $\rho(x,\bar x_{t_1,\dots,t_s})\le \eta$, as we claimed.
This implies the first statement of Lemma~C.
 
The second, more general statement follows in the same way. We only have
to observe that the number $\alpha=\alpha(\eta)$ can be chosen as the
same number for all $x\in\bold K$ in a compact set $\bold K\in
D([0,1])$. This follows from the characterization of compact sets in
the space $D([0,1])$. (See relation (14.33) in Theorem~14.3 in the book
of Billingsley~[1].)
\medskip\noindent
{\it Proof of Theorem~4.}\/ Let us construct the following coupling of
the random broken lines $\tilde S_N(\cdot,\oo)$ and $T_N(\cdot,\oo)$
which are defined with the help of the random variables $\tilde
S_k(\oo)$ and $T_k(\oo)$, $k=1,2,\dots$, in formula~(2.13). Let
$P_N^{\e,\delta}(\oo)$, $\oo\in \Omega$, be a measure on
$D([0,1])\times D([0,1])$ concentrated on the pairs, $(\tilde
S_k(\cdot,\oo),T_k(\cdot,\oo))$ in such a way that
$$
P_N^{\e,\delta}(\oo)(\tilde S_k(\cdot,\oo),T_k(\cdot,\oo))
=\mu_N(\oo)(T_k(\cdot,\oo))=\frac1{\log
\dfrac{B_N}{B_1}}\log \frac{B_{k+1}}{B_{k}},
\quad 1\le k<N.
$$
(The parameters $\e>0$ 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 $0=t_0<t_1<\cdots<t_s=1$ such that
$t_j-t_{j-1}\ge\e$, and $\supp_{t_{j-1}\le u,v<t_j}|x(u)-x(v)|\le\e$
for all $j=1,2,\dots$.
 
For all $\oo\in\Omega$ for which the sequence of probability measures
$\mu_N(\oo)$, $N=1,2,\dots$, defined by relations (2.11) and (2.12) are
compact fix a compact set $\bold K=\bold K(\e,\oo)\in D([0,1])$ in
such a way that $\mu_N(\oo)(\bold K)\ge1-\e$. We have
$\limm_{\delta\to0}w'_x(\delta)=0$ for arbitrary $x\in D([0,1])$.
Moreover, $\limm_{\delta\to0}\supp_{x\in\bold K} w'_x(\delta)=0$
for an arbitrary compact set $\bold K\subset D([0,1])$. (See,
Theorem~14.3 in Billingsley book~[1].) Given some $\delta/2>0$ 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 $0<t\le1$ define
the function $\bar x\in D([0,1])$ as $\bar x_t(u)=x(tu)$, $0\le u\le1$.
Then there exists an $\eta_1>0$ such that $d(x_t,\bar
x_t)<K(t^{-\alpha/2}-1)\le\frac\delta2$ if $x\in \bar{\bold K}$ and
$1-\eta_1\le t\le1$. Hence it is enough to show that there is some
$\eta'>0$ 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
$1-\eta_3<t\le1$ with the number $\eta_0$  appearing in relation~(3.13).
 
By Lemma~B to prove Theorem~5 it is enough to show that for an
arbitrary compact set $\bold K\subset D([0,1])$ and $\alpha>0$
$$
\liminf_{N\to\infty}(\bar\mu_N(\oo)(\bold K^\alpha)-\mu_N(\oo)(\bold
K))\ge0\quad\text{for almost all }\oo\in\Omega, \tag3.14
$$
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 $\mu'_N(\oo)$, $N=1,2,\dots$, $\mu'_N(\oo)(\bar
S_k(\cdot,\oo))=\dfrac1{\log\dfrac{B_N}{B_1}}\log\dfrac{B_{k+1}}{B_k}$,
$1\le k<N$, are convergent hence compact for almost all $\oo\in\Omega$.
Let us fix some $\e>0$. 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 $\bold K_0(N)\in\bold D([0,1])$
$$
\bold K_0(N)=\{S_k(\cdot,\oo)\: k\in \Cal C,\;1\le k<N\},\quad
N=1,2,\dots.
$$
Then $\bold K_0(N)\subset \bold K$, and
$$
\limsupp_{N\to\infty}(\mu_N(\oo)(\bold K)-\mu_N(\oo)(\bold K_0(N)))<2\e
\tag3.17
$$
for almost all $\oo\in\Omega$. To see the last relation observe that
$\limm_{N\to\infty}\mu_N(\oo)(S_k(\oo))=0$ for all fixed $k$, and
because of Theorem~5A
$$
\limm_{N\to\infty}\mu_N(\oo)\left\{\text{the union of random broken
lines }S_n(\oo)\:\rho(S_n(\cdot,\oo),\bar
S_n(\cdot,\oo))\ge \frac\alpha2\right\}=0.
$$
 
Let us define the following enlargements of the set
$\Cal C\cap\{k\: 1\le k\le N\}$:
$$
\bar{\Cal C}_N=\bar{\Cal C}_N(\alpha,\e,\oo,\bold K)=
\left\{m\:\text{there is some } n\in \Cal C, 1\le n\le N, \text{ such
that } \tilde n(n)\le m\le n\right\},
$$
$N=1,2,\dots$, and the sets of trajectories
$$
\bold K^*(N)=\left\{\bar S_m(\cdot,\oo)\: m\in\bar{ \Cal C}_N\right\}.
$$
Then $\bar{\bold K}^*(N)\subset \bold K^\alpha$ because of the
properties of the sets $\Cal C$ and $\bar{\Cal C}_N$. On the other
hand, the set $\bar{\Cal C}_N$ consists of disjoint intervals of
integers $[L_j,R_j)$ such that $\dfrac {L_j}{R_j}\ge1-\dfrac\eta3$,
and $L_j\ge (1-\eta)M$. (Here we use that if the number $M$ is chose
sufficiently large, then for all $n\ge M$ the two sides in the
defining inequalities in relation ~(3.15) are almost equal.)
 
We claim that if we choose the constant $M$ sufficiently large, then
$$
\limsup_{N\to\infty}\frac{\bar\mu_N(\oo)(\bold
K^*(N))}{\mu_N(\oo)(\bold K^*(N))}\ge1-\e \tag3.18
$$
Indeed,
$$
\frac{\bar\mu_N(\bold K^*(N))}{\mu_N(\bold K^*(N))}\ge\inf_j
\frac{\bar\mu_N\(\bigcupp_{L_j\le m< R_j} S_m(\cdot,\oo)\)}
{\mu_N\(\bigcupp_{L_j\le m< R_j} S_m(\cdot,\oo)\)}
=\inf_j\frac{\log\dfrac{B_N}{B_1}} {\log\dfrac{\bar B_N}{\bar B_1}}
\frac{\log\dfrac{\bar B_{R_j}}{\bar B_{L_j}}}
{\log\dfrac{B_{R_j}}{B_{L_j}}}, \tag3.19
$$
where $[L_j,R_j)$ are the (disjoint) intervals whose union is the set
$\bar{\Cal C}_N$. To prove (3.18) make the following observations:
Since $\limm_{N\to\infty}B_N=\infty$ and
$\limm_{N\to\infty}\dfrac{\bar B_N}{B_N}=1$, the relation
$\limm_{N\to\infty}\dfrac{\log\dfrac{B_N}{B_1}} {\log\dfrac{\bar
B_N}{\bar B_1}}=1$ holds. On the other hand
$\log\dfrac{\bar B_{R_j}}{\bar B_{L_j}}\ge \dfrac\eta4$ for
sufficiently large $M$, since the interval $[L_j,R_j)$ contains an
interval $[\tilde n(n),n]$. This inequality together with the relation
$\limm_{N\to\infty}\dfrac{\bar B_N}{B_N}=1$ imply that the inequality
$\dfrac{\bar B_{R_j}}{\bar B_{L_j}}\ge\(1-\dfrac{\e\eta}{100}\)
\dfrac{\bar B_{R_j}}{\bar B_{L_j}}$, hence $\log\dfrac
{\bar B_{R_j}}{\bar B_{L_j}}\ge\(1-\dfrac\e4\)
\log\dfrac{B_{R_j}}{B_{L_j}}$. The above estimates together with relation
(3.18) imply inequalities (3.19). Relations (3.17), (3.18) and the  relation
$\bold K^\alpha\supset\bold K^*(N)\supset \bold K_0(N)$ imply that for
sufficiently large $N$
$$
\align
\bar\mu_N(\oo)(\bold K^\alpha)&\ge \bar\mu_N(\oo)(\bold K^*(N))\ge(1-\e)
\mu_N(\oo)(\bold K^*(N))\\
&\ge(1-\e)\mu_N(\oo)(\bold K_0(N)))\ge(1-\e)\mu_N(\oo)(\bold K))-2\e.
\endalign
$$
Since this relation holds for all $\e>0$ 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