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\centerline{\bf ALMOST SURE FUNCTIONAL LIMIT THEOREMS}
\centerline{Part II. The case of independent random variables}\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 the first part of this paper we
formulated and proved the almost sure functional limit theorem under
general conditions. In this paper we prove with its help that the usual
conditions of limit theorems for the distribution of appropriately
normalized sums of independent random variables are also sufficient for
the almost sure functional limit theorem for these independent random
variables.\par}\par}
 
\beginsection 1. Introduction
 
In this paper we study the almost sure functional limit theorem for
independent random variables. To make the paper more accessible we
recall some definitions given in Part~I. of this work.
 
Let $\xi_n(\oo)$, $n=1,2,\dots$, be a sequence of random variables on a
probability space $(\Omega,\Cal A,P)$, 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 a monotone increasing sequence $B_n$,
$n=0,1,\dots$, of real numbers be given such that
$$
 B_0=0,\quad \limm_{n\to\infty}B_n=\infty, \text{ and}\quad
\limm_{n\to\infty}\dfrac{B_{n+1}}{B_n}=1 \tag1.1
$$
together with a positive number $\alpha>0$, and define, with the help of
the above partial sums $S_n(\oo)$, $n=1,2,\dots$, the broken lines
$$
\aligned
S(s,\oo)&=S_j(\oo)\quad\text{if } B_{j-1}\le s <B_j,\\
S_k(s,\oo)&=B_k^{-1/\alpha}S(B_ks,\oo), \quad 0\le s\le 1,\\
&=\frac{S_{j-1}(\oo)}{B_k^{1/\alpha}}
\quad\text{if } s_{j-1,k}\le s< s_{j,k}, \;1\le j\le k,\quad
S_k(1,\oo)=\frac {S_k(\oo)}{B_k^{1/\alpha}},\\
&\qquad\qquad\qquad\; k=1,2,\dots,
\endaligned \tag1.2
$$
where $s_{j,k}=\dfrac{B_j}{B_k}$, $0\le j\le
k$. 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
a sequence of real numbers $B_n$, $n=1,2,\dots$, be given which
satisfies formula (1.1) together with some $\alpha>0$. Let us consider
the random broken lines $S_k(s,\oo)$, $0\le s\le 1$, defined with the
help of their partial sums $S_k(\oo)=\summ_{j=1}^k\xi_j(\oo)$,
$k=1,2,\dots$, by formula (1.2). For all $\oo\in\Omega$ and
$N=1,2,\dots$ define the random probability measures $\mu_N(\oo)$
on the space $D([0,1])$ of c\`adl\`ag (continuous from
the right, limit from the left) functions on the interval $[0,1]$ in
the following way: The measure $\mu_N(\oo)$ is concentrated on the
random broken lines $S_k(\cdot,\oo)$, $1\le k\le N$, defined in
formula~(1.2), and
$$
\mu_N(\oo)(S_k(\cdot,\oo))
=\frac1{\log\dfrac{B_N}{B_1}}\log\frac{ B_{k+1}}{B_k},\quad 1\le k< N.
\tag1.3
$$
The sequence of random variables $\xi_n(\oo)$, $n=1,2,\dots$, satisfies
the almost sure functional limit theorem with weight function $B_n$,
$n=1,2,\dots$, parameter $\alpha>0$ and limit measure $\mu_0$ on the
space $D([0,1])$ if for almost all $\oo\in\Omega$ the probability
measures $\mu_N(\oo)$ defined with the help of the above constants
$B_n$ and $\alpha$ (appearing in formula (1.2)) converge weakly to the
measure $\mu_0$ as $N\to\infty$.} \medskip
This definition can be naturally modified to measures in the space
$C([0,1])$ of continuous functions on the interval $[0,1]$.
It follows from the general theory that if the almost sure functional
limit theorem holds in the space $D([0,1])$ and the limit measure
$\mu_0$ is concentrated in the space $C([0,1])$, then the $C([0,1])$
version of the almost sure functional limit theorem also holds. We
formulated the almost sure functional limit theorem in the $D([0,1])$
space, because we want to prove it also for random variables in the
domain of attraction of a stable law. In this case we have to work in
the space $D([0,1])$.
 
The definition of the almost sure functional limit theorem given here
slightly differs from that given in Part~I. of this paper (see~[18].)
In the definition given there we have considered a sequence
of $A_n$ instead of the number $\alpha$. But since in all cases we
prove the almost sure functional limit theorem a sequence of the form
$A_n=B_n^{1/\alpha}$ is chosen, we made this modification. In Section~2
of Part~I. of this paper we formulated and proved a Corollary
which states the following: Let a self-similar process $X(t,\oo)$,
$t\ge0$, be given with a self-similarity parameter $\alpha>0$, (see
its definition in Part~I.) which also satisfies some additional weak
conditions which are not serious restriction in possible applications,
together with a sequence $B_n$, $n=1,2,\dots$, of real numbers for
which relation (1.1) holds. Then the random variables
$\eta_n(\oo)=X(B_n,\oo)-X(B_{n-1},\oo)$, $n=1,2,\dots$, satisfy the
almost sure functional limit theorem with weight function $B_n$,
$n=1,2,\dots$, parameter $\alpha$ and limit measure $\mu_0$ which is
the distribution of the self-similar process restricted to the
interval~$[0,1]$.
 
In particular, by applying this result with the choice of the Wiener
process $W(t,\oo)$ as the self-similar process together with a
sequence $B_n$ satisfying formula (1.1) we get that the almost sure
functional limit theorem holds for a sequence of independent Gaussian
random variables $\eta_n(\oo)$, $n=1,2,\dots$, $E\eta_n(\oo)=0$,
$E\eta_n^2(\oo)=B_n-B_{n-1}$ with the weight functions $B_n$, parameter
$\alpha=2$ and limit measure $\mu_0$ which is the Wiener measure, i.e.\
the distribution of the process $W(t,\oo)$, $0\le t\le1$. Similarly, we
get by considering a stable process $X(t,\oo)$ with self-similarity
parameter $\alpha$, $0<\alpha<2$, $\alpha\neq1$, as a self-similar
process and a sequence of real numbers $B_n$ satisfying relation (1.1),
that a sequence of independent random variables $\eta_n(\oo)$,
$n=1,2,\dots$, such that the distribution of $\eta_n(\oo)$ agrees with
the distribution of $X(B_n-B_{n-1},\oo)$ satisfies the almost sure
functional limit theorem with weight function $B_n$, parameter $\alpha$
and limit measure $\mu_0$ which is the distribution of the restriction
of the process $X(t,\oo)$ to the interval $[0,1]$. We prove that if a
sequence of independent random variables is given whose normalized
partial sums converge in distribution either to the standard normal
distribution or to a stable law, then this sequence also satisfies the
almost sure functional limit theorem. Such statements are the main
results of this paper. They are formulated in the following Theorems.
\medskip\noindent
{\bf Theorem 1.} {\it Let $\xi_n(\oo)$, $n=1,2,\dots$, be a sequence of
independent random variables such that $E\xi_n(\oo)=0$,
$E\xi_n^2(\oo)=\sigma_n^2$, and it satisfies the Lindeberg condition,
i.e.
$$
\lim_{n\to\infty}
\frac1{D_n^2}\sum_{k=1}^nE\xi_k^2(\oo)I(|\xi_k(\oo)|\ge
\e D_n)=0\quad\text{for all }\e>0, \tag1.4
$$
where $D_n^2=\summ_{k=1}^n\sigma_k^2$. Then the sequence of random
variables $\xi_n(\oo)$, $n=1,2,\dots$, satisfies the almost sure
functional (central) limit theorem with the Wiener measure $\mu_0$ as
the limit measure, weight function $B_n=D_n^2$, $n=1,2,\dots$, and
parameter~$\alpha=2$.} \medskip\noindent
{\bf Theorem 2.} {\it Let $\xi_n(\oo)$, $n=1,2,\dots$, be a sequence
of independent, identically distributed random variables with a
non-degenerated distribution such that $E\xi_1(\oo)=0$ and
$\mu(x)=E\xi_1^2(\oo)I(|\xi_1(\oo)|\le x)$ is a slowly varying
function at infinity. Define the numbers $a_n$ as
$a_n=\sup\left\{u\:n\dfrac{\mu(u)}{u^2}\ge1\right\}$
for all~$n\ge n_0$ with a sufficiently large integer~$n_0$. (This
definition is meaningful. To see this observe that
$n\dfrac{\mu(u)}{u^2}>1$ for an appropriate $u$ and all
sufficiently large $n$, and $\limm_{u\to\infty}\dfrac{n\mu(u)}{u^2}=0$
for all $n$ if $\mu(u)$ is a slowly varying function.) Define the
sequence $a_n$ in the above way if $n\ge n_0$, and for the sake of a
unique definition put $a_n=a_{n_0}$ for $n\le n_0$. Then the sequence
$\xi_n(\oo)$, $n=1,2,\dots$, satisfies the almost sure (central)
functional limit theorem with the Wiener measure $\mu_0$ as the limit
measure, weight function $B_n=\summ_{k=1}^n\mu(a_k)$, $n=1,2,\dots$, and
parameter~$\alpha=2$.}\medskip
 
To formulate the following Theorem~3 let us recall that the distribution
function $G(x)$ of a stable law, $0<\alpha<2$, is determined by three
parameters $\alpha$, $C_1\ge0$ and $C_2\ge0$, $C_1+C_2>0$, such that
the relations
$$
\aligned
G(x)&\sim C_1 x^{-\alpha}\quad\text{if } x\to\infty, \\
1-G(-x)&\sim C_2 x^{-\alpha}\quad\text{if } x\to\infty
\endaligned \tag1.5
$$
hold. Let us also recall that for all stable laws $G(x)$ satisfying
(1.5) with $0<\alpha<2$, $\alpha\neq1$, there is a (stable) process
$X(t,\oo)$, $0\le t\le1$, with trajectories in the space
$D([0,1])$ such that it has independent and stationary increments,
and $X(1,\oo)$ has the distribution function $G(x)$. The distribution of
this process $X(t,\oo)$ in the space $D([0,1])$ is uniquely determined.
It is a self-similar process with self-similarity parameter~$\alpha$.
 
In the next Theorem 3 we formulate the following result: If such
conditons are imposed under which the normalized partial sums of a
sequence of independent, identically distributed random converge in
distribution to a stable law, then these random variables also satisfy
the almost sure functional limit theorem. Before formulating the
precise statement let us introduce some notations. Let us consider a
distribution function $F(x)$ which satisfies the following condition:
$$
\aligned
1-F(x)&\sim C_1x^{-\alpha}L(x)\\
F(-x)&\sim C_2x^{-\alpha}L(x)
\endaligned \qquad\text{if }x\to\infty, \tag1.6
$$
where $L(x)$ is a slowly varying function at infinity,
$C_1\ge0$, $C_2\ge0$, $C_1+C_2>0$, $0<\alpha<2$, $\alpha\neq1$. Let us
define the following functions $b(x)$ and $\bar L(x)$ which appear in
the normalization of the almost  sure functional theorem (and
also in the usual limit theorem) for i.i.d.\ random variables with
distribution function $F(x)$ which satisfies condition (1.6). For all
$x>0$ put $b(x)=\max\left\{u\:\dfrac{L(u)x}{u^\alpha}\ge1\right\}$, and
let $\bar L(x)=b(x)^{\alpha}x^{-1}$, where the number $\alpha$ and
function $L(\cdot)$ are the same as in formula (1.6). Let $\mu_0$ denote
the (uniquely determined) distribution of the stable process
$X(t,\oo)$ with parameter $\alpha>0$, $0\le t\le 1$, in the space
$D([0,1])$ for which $X(1,\oo)$ has that stable distribution $G(x)$
which satisfies relation~(1.5). (The numbers $\alpha$, $C_1$ and $C_2$
are the same in formulas (1.5) and (1.6).)
\medskip\noindent
{\bf Theorem 3.} {\it Let $\xi_n(\oo)$, $n=1,2,\dots$, be a sequence of
independent and identically distributed random variables with a
distribution function $F(x)$ which satisfies Condition (1.6).
Then there is a sequence of real numbers $a_n$ such that the sequence of
random variables $\xi_n(\oo)-a_n$, $n=1,2,\dots$, satisfies the almost
sure functional limit theorem with weight function $B_n=\summ_{k=1}^n
\bar L(k)$, $n=1,2,\dots$ and limit measure $\mu_0$, where the function
$\bar L(x)$ and measure $\mu_0$ were defined before the formulation of
this Theorem~3.
 
We also claim that the function $b(x)$ defined before the formulation
of this Theorem~3 is a regularly varying function at infinity with
parameter $1/\alpha$, hence $\bar L(x)=b(x)^{\alpha}x^{-1}$ is a slowly
varying function.
 
The normalized partial sums $\dfrac1{B_n^{1/\alpha}}\summ_{k=1}^n
(\xi_k(\oo)-a_n)$  converge in distribution to $G(x)$ as $n\to\infty$
if the constants $a_n$ are the same as in the almost sure functional
limit theorem formulated in this theorem.} \medskip\noindent
{\it Remark 1:}\/ With some modification in the proof the last
statement of Theorem~3, the limit theorem for the distribution of the
normalized partial sums, can be replaced by a stronger version of this
result. The functional limit theorem also holds for the distributions
of the random broken lines made for all $n=1,2,\dots$ from the
normalized partial sum $S_k(\oo)=\dfrac1{B_n^{1/\alpha}}\summ_{j=1}^k
(\xi_j(\oo)-a_n)$, $k=1,\dots,n$, in the natural way. The limit is the
same measure $\mu_0$ which appears as the limit in the almost sure limit
theorem. \medskip
We also explain without working out all details that Theorem~3
also holds for $\alpha=1$ with some slight natural
modifications.\medskip\noindent
{\bf Theorem 3$'$.} {\it The results of Theorem~3 also hold in the case
$\alpha=1$ with the slight modification that in this case the ``shift
parameters" $a_n$, $n=1,2,\dots$ must be defined differently, the
measure $\mu_0$ is the distribution of the process $X(t)-\gamma t\log t$
with the constant $\gamma=\gamma(C_1,C_2)=C_1-C_2$, where $X(t,\oo)$ is
the stable process with parameter $\alpha=1$ for which $X(1,\oo)$ has
distribution function $G(x)$ which satisfies relation (1.5).}
\medskip\noindent
{\it Remark 2.}\/ The norming (shift) constants $a_n$ in the limit
theorem for the distribution of the normalized partial sums
mentioned at the end of  Theorem~3 are determined with a certain
accuracy uniquely. Namely, the norming constants $a_n$ can be replaced
by another norming constant $\bar a_n$ if and only if $a_n-\bar
a_n=o\(\dfrac{B_n^{1/\alpha}}n\)$.  This implies that the norming
sequence $a_k$, $k=1,2,\dots$, in the almost sure functional limit
theorem  can be replaced by any such sequence $\bar a_k$ for which the
number $\bar a_n$ can be chosen as the norming (shift) constant in the
partial sum of the first $n$ term in the limit theorem for the
distribution of the partial sums. To see this it is enough to observe
that $\limm_{n\to\infty}\summ_{k=1}^n\dfrac{a_k-\bar a_k}
{B_n^{1/\alpha}}=0$, since  $\summ_{k=1}^n\dfrac{a_k-\bar a_k}
{B_n^{1/\alpha}}=o\(\summ_{k=1}^n\dfrac{B^{1/\alpha}_k}
{kB_n^{1/\alpha}}\)=o(1)$. \medskip
 
Let us remark that the conditions imposed in Theorems~1 and~2 are
the natural conditions for the central limit theorem for sums of
independent random variables. Theorem~1 contains the necessary and
sufficient conditions of the central limit theorem for sums of
independent random variables with the natural norming constants
$D_n^{1/2}$  if these random variables satisfy the condition of uniform
smallness. The conditions in Theorem~2 are the necessary and sufficient
conditions of the central limit theorem with an appropriate
normalization for sums of independent and identically distributed
random variables. Similarly, in Theorem~3 the necessary and sufficient
conditions of the limit theorem with a stable limit law for partial
sums of independent and identically distributed random variables were
imposed.
 
The above results will be proved by means of a coupling argument. In
Part~I. we introduced a notion we called the Property~A. We showed
that if Property~A holds for a pair of sequences of 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 of random variables $\xi_n(\oo)$,
$n=1,2,\dots$, also satisfies the almost sure functional limit theorem
with the same weight function $B_n$, parameter $\alpha$ and limit
measure $\mu_0$ as the sequence $\eta_n(\oo)$, $n=1,2,\dots$. We shall
prove Theorems 1,~2 and~3 with the help of this result which we recall
in the next section.
 
A result we shall call the Basic Lemma will be formulated and proved.
Then it will be proved with its help that the sequences of random
variables considered in these theorems together with a sequence of
appropriately defined independent random variables with normal or
stable distributions satisfy Property~A. The theorems follow from
Property~A for these pairs of sequences and the almost sure functional
limit theorem for independent random variables with Gaussian or stable
distribution mentioned in the beginning of this paper.
 
This paper consists of six sections. In Section~2 we formulate a result
we call the Basic Lemma. This Basic Lemma will be proved in Section~3.
In Section~4 we prove Theorems~1 and~2, some almost sure functional
limit theorems with the Wiener measure as the limit measure. In Section
5 we prove Theorem~3, the almost sure functional limit theorem for
independent, identically distributed random variables in the domain of
attraction of a stable law. Finally, Section~6 contains the formulation
of certain open problems and some comments. Here we also compare
briefly our results with those of earlier papers.
 
\beginsection 2. Formulation of Property A and the Basic Lemma
 
First we recall the definition of Property~A which enables us to prove
the  almost sure functional limit theorem in several interesting cases.
\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
$D([0,1])$ with some weight function $B_n$, $n=0,1,\dots$, satisfying
relation (1.1) and parameter $\alpha>0$. 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 $0\le u\le1$.
 
Define the indices $N(n)$ as $N(n)=\inf\{k\: B_k\ge 2^n\}$,
$n=0,1,\dots$. The pair of sequences of random variables
$(\xi_n(\oo),\eta_n(\oo))$, $n=1,2,\dots$, satisfies Property~A  if for
all $\e>0$ there exists a sequence of random variables
$\tilde\xi_n(\oo)=\tilde\xi_n(\e,\oo)$, $n=1,2,\dots$, 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
$T_n(\oo)=\summ_{k=1}^n\eta_k(\oo)$, $n=1,2,\dots$, satisfy the relation
$$
\limsup_{n\to\infty}\frac1n\sum_{k=1}^{N(n)}\log\frac
{B_{k+1}}{B_{k}} I\(\left\{\oo\:\frac{\supp_{0\le
j\le k}|\tilde S_j(\oo)-T_j(\oo)|}{B_k^{1/\alpha}}>\e\right\}\)\le
\e \tag2.1
$$
for almost all $\oo\in\Omega$, where $I(A)$ denotes the indicator
function of the set $A$.}\medskip
In Theorem 4 of Part I. we have proved that if the pair of sequences
of random variables $(\xi_n(\oo),\eta_n(\oo))$, $n=1,2,\dots$, satisfies
Property A, then the sequence of random variables $\xi_n(\oo)$,
$n=1,2,\dots$, satisfies the almost sure functional limit theorem with
the same weight function $B_n$, parameter $\alpha>0$ and limit measure
$\mu_0$ as the sequence of random variables $\eta_n(\oo)$,
$n=1,2,\dots$. In this section we formulate a lemma which enables us to
check Property~A in several interesting cases.
 
To prove Property~A we need a good construction of the pairs
$(\tilde\xi_n(\oo),\eta_n(\oo))$, $n=1,2,\dots$. Let us first briefly
describe a standard method which produces a construction for partial
sums of independent random variables in such a way that the differences
of the partial sums $\tilde S_n(\oo)-T_n(\oo)$, $n=1,2,\dots$, made from
these random variables are relatively small for almost all $\oo$.
 
Let us choose an appropriate subsequence $n_j$ of the integers and apply
a so-called quantile transform, to be described later, which makes the
differences $(S_{n_j}(\oo)-S_{n_{j-1}}(\oo))-(T_{n_j}(\oo)-
T_{n_{j-1}}(\oo))$ relatively small, and for which the expressions
$S_{n_j}(\oo)-S_{n_{j-1}}(\oo)$ and $T_{n_j}(\oo)-T_{n_{j-1}}(\oo)$
have the right distribution. We make such constructions independently
for all $j=1,2,\dots$, and define the random variables $S_{n_j}(\oo)$
and $T_{n_j}(\oo)$ as the partial sums of these terms. Then these
subsequences can be extended to two sequences $S_n(\oo)$ and $T_n(\oo)$
which have the same joint distribution as the partial sums of the
independent random variables $\xi_k(\oo)$  and $\eta_k(\oo)$,
$k=1,2,\dots$. The differences between the random variables
$S_{n_j}(\oo)-S_{n_{j-1}}(\oo)$ and $T_{n_j}(\oo)-T_{n_{j-1}}(\oo)$
constructed in the above way can be well estimated if we have
a good control on the distance of the distribution functions of these
partial sums. This enables us to bound the differences
$S_{n_j}(\oo)-T_{n_j}(\oo)$, and then by bounding the fluctuation of
the sequences $S_n(\oo)$ and $T_n(\oo)$ between these points we get an
estimate about the goodness of this approximation.
 
Such a construction, with the appropriate choice of the numbers $n_j$ is
made in certain papers to get an almost sure approximation of a sequence
$T_n(\oo)$ of partial sums of independent random variables with a
sequence $S_n(\oo)$, $n=1,2,\dots$, of partial sums of different
independent random variables. The construction we make to satisfy
Property~A in case of independent random variables is similar. The only
difference is that we want to get a good bound on the expression in
formula (2.1) instead of an almost sure approximation, hence we choose
the sequence $n_j$ according to this requirement. Now we only demand
that the differences $S_n(\oo)-T_n(\oo)$ be small for most indices $n$.
The existence of some exceptional indices $n$ (depending on $\oo$)
where this difference is large is allowed if they do not enlarge
considerably the expression in formula~(2.1). The role of the following
Basic Lemma is that it enables us to show that a construction
satisfying Property~A can be made under relatively weak conditions.
Before its formulation we give an informal explanation about the
technical details n it, and also make some indication about its role in
the proof of Theorems~1---3.
 
The Basic Lemma actually states that the above sketched construction
with an appropriate choice of the points $n_j$ satisfies  inequality
(2.1). In this lemma we give a bound for the partial sums of some random
variables $\zeta_k(\oo)$, $k=1,2,\dots$, if they satisfy certain
conditions. The bound (2.6) proved in the Basic Lemma will be
applied with the choice $\zeta_k(\oo)=\tilde\xi_k(\oo)-\eta_k(\oo)$
where $\tilde\xi_k(\oo)$ and $\eta_k(\oo)$, $k=1,2,\dots$, are random
variables constructed in the above way. Let us remark that when the
sequences $S_n(\oo)$ and $T_n(\oo)$ are compared, then the natural
time scale is measured by the sequence $B_n$. (To understand this let us
look at the definition of the process $S(t,\oo)$ in formula (1.2).)
Let us consider an exponentially rare sequence of the time parameter.
This is the content of the definition of the numbers $N(n)$ in the
formulation of the Basic Lemma which guarantees that $B_{N(n)}\sim2^n$.
We shall also define a refinement $N(n,k)$ of this sequence, and
these points $N(n,k)$ will play the role of the points $n_j$ where the
quantile transform will be applied in the above sketched construction.
 
We make a decomposition of the random variables $\zeta_k(\oo)$ in
formula (2.3) of the Basic Lemma which will be satisfied in our
applications with the natural choice
$\zeta^{(1)}_k(\oo)=\tilde\xi_k(\oo)$ and
$\zeta^{(2)}_k(\oo)=\eta_k(\oo)$. The Basic Lemma also contains a
condition about the independence of the random variables
$U_{N(n,k)}(\oo)-U_{N(n,k-1)}(\oo)$, $n=1,2,\dots$, $k=1,\dots,l_n$,
defined there, but this is a condition which is automatically
satisfied in the constructions we apply in the proof of Theorems 1---3.
In formula (2.5) we formulate an estimate which can be satisfied in our
applications if we define the points $N(n,k)$ in an appropriate way
and give a good estimate for the difference of the partial sums of the
random variables $\tilde\xi_j(\oo)$ and $\eta_j(\oo)$ with indices $j$
between these points by means of an estimate on the quantile transform.
In our applications formula (2.4) states a good bound for the
fluctuation of the partial sums of the random variables
$\xi_k(\oo)$ and $\eta_k(\oo)$, $k=1,2,\dots$. Let us also remark we
need the bounds in formulas (2.4) and (2.5) only for large indices $n$,
and the threshold index from which they must hold may depend on the
parameter $\e>0$ appearing in formula (2.6). These are the conditions
imposed in the Basic Lemma to satisfy formula (2.6).
 
We shall prove Theorems 1 --- 3 by means of the Basic Lemma.
An important step of the proof is a good choice of the points
$N(n,k)$ between which the quantile transform will be applied. This
choice is essentially different in the proof of Theorem~3 and Theorem~1.
In Theorem~3 the points $N(n,k)$ contain all integer points between
$N(n-1)$ and $N(n)$. In this case formula (2.4) is an empty
condition. In the proof of Theorem~1 the numbers $N(n,k)$ will be chosen
in such a way that $B_{N(n,k+1)}-B_{N(n,k)}\sim \bar \e2^n$, where the
coefficient $\bar\e>0$ is a very small number, but it does not depend on
the number $n$. This means that the numbers $N(n,k)$ (with fixed~$n$)
are relatively uniformly distributed in the interval $[N(n-1),N(n))$,
and the difference between them is relatively large. The cause of the
different choice of $N(n,k)$ in the proof of Theorem~1 and Theorem~3 is
that under the conditions of Theorem~3 the single terms $\xi_n(\oo)$ and
$\eta_n(\oo)$ have a similar distribution, while under the conditions
of Theorem~1 one can guarantee the similar distribution of the partial
sums of the random variables $\xi_n(\oo)$ and $\eta_n(\oo)$, by means of
the central limit theorem, only if these partial sums have sufficiently
many terms. In the proof of Theorem~2 the single terms $\xi_n(\oo)$
will be written up as sums of random variables by means of an
appropriate truncation. Then in the proof of Theorem~2 different
partial sums have to be handled. All of them will be investigated by
means of the Basic Lemma, but some of them will be estimated with a
choice of the numbers $N(n,k)$ similar to that given in the proof of
Theorem~1 and some of them with a choice of the numbers $N(n,k)$
similar to that given in the proof of Theorem~3.
 
Now we turn to the formulation of the Basic Lemma. First we introduce
the following definition. \medskip\noindent
{\bf Definition of refinement of a sequence of integers.} {\it Given a
sequence  $0=N(0)<N(1)<N(2)<\cdots$ of integers we call  the refinement
of this sequence $N(n)$, $n=0,1,2,\dots$ a set of non-negative integers
$N(n,k)$ indexed by two parameters $n=1,2,\dots$, and $0\le k\le l_n$
with some positive integer $l_n$ such that
$$
N(n-1)=N(n,0)<N(n,1)<\cdots<N(n,l_n)=N(n),\quad n=1,2,\dots.
$$
}\medskip\noindent
{\bf Basic Lemma.} {\it Let $B_n$, $n=0,1,\dots$, $B_0=1$, be a
sequence of real numbers which satisfies relation (1.1). Let a
sequence of random variables $\zeta_n(\oo)$, $n=1,2,\dots$ and a number
$\alpha>0$ also be given. Define the sequence $N(n)=\inf\{k\: B_k\ge
2^n\}$, $n=1,2,\dots$, $N(0)=0$. We give an estimate on the maximum of
the partial sums of the random variables $\zeta_n(\oo)$ under
appropriate conditions.
 
Fix a number $\e>0$ and a refinement $N(n,k)$, $n=0,1,\dots$, $0\le k\le
l_n$, of the sequence $N(n)$, $n=1,2,\dots$, which may depend on $\e$.
Put $$
\align
U_n(\oo)&=\summ_{j=1}^n\zeta_j(\oo),\quad n=1,2,\dots,\\
\intertext{and}
V_{n}(\oo)&=\supp_{0<j\le l_n}\left|U_{N(n,j)}(\oo)-
U_{N(n,0)}(\oo)\right|, \quad  n=1,2,\dots.
\tag2.2 \endalign
$$
Let us assume that there is a decomposition
$$
\zeta_k(\oo)=\zeta^{(1)}_k(\oo)-\zeta_k^{(2)}(\oo),\quad k=1,2,\dots
\tag2.3
$$
of the random variables $\zeta_k(\oo)$ in such a way that both sequences
$\zeta_k^{(i)}(\oo)$, $k=1,2,\dots$, $i=1,2$, consist of independent
random variables which satisfy certain inequalities formulated below. To
formulate them let us introduce the notation
$$
\align
\zeta^{(i)}_{n,k,m}(\oo)&=\zeta_{N(n,k-1)+m}^{(i)}(\oo), \quad
n=1,2,\dots,\; k=1,\dots,l_n,\\
&\qquad 1\le m\le N(n,k)-N(n,k-1),\quad i=1,2
\endalign
$$
with the help of the refinement $N(n,k)$ of the sequence $N(n)$,
$n=1,2,\dots$, fixed in this lemma. Let us assume that the following
inequalities hold:
$$
\aligned
P&\(\sup_{1\le p<N(n,k)-N(n,k-1)}\left|\sum_{m=1}^p
\zeta^{(i)}_{n,k,m}(\oo)\right| >\e x2^{n/\alpha}\)\le \frac{C_1 \e
x^{-\gamma}}{l_n} \\
&\qquad\qquad i=1,2,\quad\text{for all }x\ge 1,\;\;n\ge n_0,\text{
and }1\le k\le l_n \endaligned \tag2.4
$$
with some constants $n_0=n_0(\e)>0$, $\gamma>0$ and $C_1>0$. (In the
case $N(n,k)=N(n,k-1)+1$ this sum is empty. In this case we assume that
relation (2.4) is satisfied.) Let us also assume that
the random variables $U_{N(n,k)}(\oo)-U_{N(n,k-1)}(\oo)$, $n=1,2,\dots$,
$k=1,\dots,l_n$, are independent, and the inequality
$$
P\(V_n(\oo)\ge \e x2^{n/\alpha}\)\le C_2 \e x^{-\gamma}\quad\text {for
all } x\ge 1 \text{ and } n\ge n_0 \tag2.5
$$
holds with some $n_0=n_0(\e)$, $\gamma>0$  and $C_2>0$. Then
$$
\aligned
\limsup_{n\to\infty}\frac1n\sum_{k=1}^{N(n)}\log\frac
{B_{k+1}}{B_{k}} I&\(\frac{\supp_{0\le s\le
k}|U_s(\oo)|}{B_k^{1/\alpha}}> K\e\)\le K\e \\
&\qquad\qquad\qquad\text{for almost all } \oo\in\Omega.
\endaligned \tag2.6
$$
with an appropriate constant $K=K(C_1,C_2,\gamma,\alpha)>0$. (Such a
constant~$K$ could be given explicitly, but we do not need such a
formula. It is enough to know that this constant $K$ does not
depend on~$\e$.)} \medskip
 
We shall be able to prove Property~A under weak conditions by applying
the Basic Lemma for arbitrary small~$\e>0$. In such a way we show
that the conditions sufficient for a limit theorem for partial sums of
independent random variables also imply Property~A with an appropriate
construction. On the other hand, a construction by means of an almost
sure approximation only supplies a weaker result. Indeed, by applying
such a construction we get bounds sufficient for our purposes only
under some additional conditions. The reason for this difference is
that the condition of Property~A formulated in formula~(2.1) only
demands that the differences $S_n(\oo)-T_n(\oo)$ be small in some
average. Let us remark that there are even results, (see Berkes and
Cs\'aki~[4]) which state that there are cases when the almost sure
functional limit theorem holds for a sequence of independent random
variables, but their partial sums do not satisfy a limit theorem.
 
Finally we remark that in Conditions (2.4) and (2.5) an estimate on the
tail behaviour of the partial sums was formulated. In a limit theorem
for normalized partial sums of independent random variables we do not
require such an estimate. But the conditions formulated in Theorems~1,
2 and~3, i.e.\ the necessary and sufficient conditions of certain limit
theorems also imply an estimate on the tail behaviour appropriate for
our purposes. Actually, conditions (2.4) and (2.5) can be considerably
weakened. The power $|x|^{-\gamma}$ at the right-hand side of these
formulas could be replaced by $(1+\log |x|)^{-\gamma'}$ with a
sufficiently large $\gamma'>0$. But such a condition does not seem to
be better applicable in the problems we are interested in.
 
\beginsection 3. Proof of the Basic Lemma
 
{\it Proof of the Basic Lemma.}\/ We shall prove relation (2.6) with
the help of two inequalities. In these inequalities the supremum is
taken for an appropriate subsequence. To formulate them we define the
numbers $L_0=1$, $L_p=\summ_{k=1}^p l_k$, $p=1,2,\dots$, and the
sequence $m(j)$, $j=0,1,\dots$, by the formula $m(0)=0$ and
$m(j)=N(p-1,j-L_{p-1})$, if $L_{p-1}<j\le L_p$. The numbers $l_p$ and
$N(p,j)$ in these formulas are the same as those considered in the
formulation of the Basic Lemma. The number $m(j)$ counts the value of
the $j$-th term among the numbers $N(n,k)$. Observe that in particular
$m(L_n)=N(n-1,l_n)=N(n)$, $n=1,2,\dots$. Put $A_n=B_n^{1/\alpha}$. We
shall prove that
$$
\aligned
\limsup_{n\to\infty}\frac1n\sum_{j=1}^{L_{n}} \log\frac{B_{m(j)+1}}
{B_{m(j-1)+1}} I&\(\frac{\supp_{1\le
s\le j}|U_{m(s)}(\oo)|}{A_{m(j)}}>K_1 \e\)\le K_1\e\\
&\qquad\qquad\qquad\qquad\text{for almost all } \oo\in\Omega
\endaligned \tag3.1
$$
with an appropriate constant $K_1>0$, and
$$
\aligned
\limsup_{n\to\infty}\frac1n \sum_{j=1}^{L_{n}}\log\frac
{B_{m(j)+1}}{B_{m(j-1)+1}} I&\(\frac{\supp_{1\le
s\le j}\supp_{m(s-1)<p<m(s)}|U_p(\oo)-U_{m(s-1)}(\oo)|}{A_{m(j)}}
>K_2 \e\)\\
&\qquad\qquad\qquad \le K_2\e \quad \text{for almost all } \oo\in\Omega
\endaligned \tag3.2
$$
with an appropriate constant $K_2>0$. First we show that relations
(3.1) and (3.2) imply relation (2.6) with $K=3^{1/\alpha}(K_1+K_2)$.
 
Indeed, if for some $\oo\in\Omega$ there is an index $k$, $n_0\le
k\le N(n)$ with some $n_0=n_0(\e)$, such that it gives a non-zero
contribution to the sum in (2.6) with the choice
$K=3^{1/\alpha}(K_1+K_2)$, i.e. $\supp_{1\le s\le
k}|U_s(\oo)|>3^{1/\alpha}(K_1+K_2)A_k\e$, then consider that interval
$(m(j-1),m(j)]$, $1\le j\le L_n$, which contains this number $k$. In
this case one of the following relations holds. Either
$$
\align
&\supp_{1\le s\le j}|U_{m(s)}(\oo)|>3^{1/\alpha}K_1A_k\e\ge
K_1A_{m(j)}\e\\ \intertext{or}
\supp_{1\le s\le j}&\supp_{m(s-1)<p<m(s)}
|U_p(\oo)-U_{m(s-1)}(\oo)|>3^{1/\alpha}K_2A_k\e\ge K_2A_{m(j)}\e.
\endalign
$$
Then the contribution of the terms with indices in the interval
$(m(j-1),m(j)]$ to the sum in the expression (2.6) is not greater
than $\log\dfrac{B_{m(j)+1}}{B_{m(j-1)+1}}$, and such a contribution
appears in the $j$-th term of one of the sum (3.1) or (3.2).
Hence relations (3.1) and (3.2), the identity $m(L_n)=N(n)$ together
with a summation for $1\le j\le L_n$ imply formula~(2.6).
 
To prove relation (3.1) introduce the random variables
$$
T_s(\oo)=U_{m(s)}(\oo)-U_{m(s-1)}(\oo), \quad s=1,2,\dots.
$$
The random variables $T_s(\oo)$, $s=1,2,\dots$ are independent. This
statement is equivalent to the independence of the random variables
$U_{N(n,k)}(\oo)-U_{N(n,k-1)}(\oo)$, $n=1,2,\dots$, $k=1,\dots,l_n$,
and this is a condition imposed in the formulation of the Basic Lemma.
 
Since $\limm_{n\to\infty}\dfrac{B_{N(n+1)}}{B_{N(n)}}=2$,
$A_n=B_n^{1/\alpha}$, there is some $n_0>0$ such that
$$
A_{N(n)}\ge 2^{(n-k)/2\alpha}A_{N(k)}\quad\text{for arbitrary }
n\ge n_0 \text{ and }k\le n.
$$
For all $s=1,2,\dots$ define the number $R(s)$ which satisfies the
inequality $L_{R(s)-1}< s\le L_{R(s)}$. The  number $R(s)$ counts the
number of the form $N(l,0)=N(l-1)$ among the first $s$ terms of the
sequence $N(n,k)$. This fact and the content of the value of $m(j)$
imply that $N(R(s)-1))< m(s)\le N(R(s))$. (The sequence $R(s)$ is the
``inverse" of the monotone sequence $L_s$. The relation $R(L_s)=s$
holds.) Hence
$$
\frac {A_{m(j)}}{A_{m(s)}}\ge
2^{(R(j)-R(s)-1)/2\alpha}\quad\text{for
}1\le s\le j\text { and }j\ge n_0.
$$
Let us fix some $j\ge n_0$. We shall show by applying the above relation
for $s\le j$ and by putting in one block those indices $s$ for which
$N(r-1)<m(s)\le N(r)$, or equivalently $L_{r-1}<s\le L_r$ that
$$
\aligned
&\left\{\oo\:\frac{\supp_{1\le s\le
j}\left|U_{m(s)}(\oo)\right|}{A_{m(j)}} \ge K_1\e\right\}
=\left\{\oo\:\left|\summ_{p=1}^s T_p(\oo)\right|\ge K_1\e
A_{m(j)} \;\text{ for some }1\le s\le j\right\}\\
&\qquad \subset\bigcup_{r=1}^{R(j)} \left\{\oo\:\sup_{L_{r-1}<u\le L_r}
\left|\sum_{p=L_{r-1}+1}^u T_p(\oo)\right| \ge
CK_1\e2^{(R(j)-r)/4\alpha}A_{m(L_{r-1})}\right\}
\endaligned \tag3.3
$$
with $C=1-2^{-1/4\alpha}$. To prove relation (3.3) observe that if
$L_{r-1}<j$, then
$$
A_{m(j)}\ge2^{(R(j)-R(L_{r-1})-1)/2\alpha}
A_{m(L_{r-1})}=2^{(R(j)-r)/2\alpha}A_{m(L_{r-1})},
$$
hence if some $\oo$ is not contained in the set at the right-hand side,
i.e.
$$
\sup_{L_{r-1}<u\le L_r}\left|\sum_{p=L_{r-1}+1}^u T_p(\oo)\right| <
CK_1\e2^{(R(j)-r)/4\alpha}A_{m(L_{r-1})}\quad
\text{for all }1\le r\le R(j)
$$
then
$$
\align
\left|\sum_{p=1}^s T_p(\oo)\right| &<\sum_{r=1}^{R(j)}
\sup_{L_{r-1}<u\le L_r}\left|\sum_{p=L_{r-1}+1}^u T_p(\oo)\right|\\
&\le CK_1\e\sum_{r=1}^{R(j)}2^{-(R(j)-r)/4\alpha}
2^{(R(j)-r)/2\alpha}A_{m(L_{r-1})} \\
&\le CK_1\e \sum_{r=1}^{R(j)}2^{-(R(j)-r)/4\alpha}A_{m(j)}<K_1\e
A_{m(j)} \endalign
$$
for all $s\le L_{R(j)}$, hence for all $s\le j$, and this means that
$\oo$ is not contained in the set  at the left-hand side of formula
(3.3).
 
We get from formula (3.3), the definition of the random variables
$V_r(\oo)$ introduced in formula (2.2) and the relation
$A_{m(L_{r-1})}=B_{m(L_{r-1})}^{1/\alpha}=
B_{N(r-1)}^{1/\alpha}\ge2^{(r-1)/\alpha}$ that
$$
\align
\left\{\oo\: \frac{\supp_{1\le s\le
j}\left|U_{m(s)}(\oo)\right|}{A_{m(j)}}\ge
K_1\e\right\} &\subset \bigcup_{r=1}^{R(j)}\left\{\oo\:
V_r(\oo)
\ge CK_1\e2^{(R(j)-r)/4\alpha}\times 2^{(r-1)/\alpha}\right\}  \\
&= \bigcup_{r=1}^{R(j)} \left\{\oo\: V_r(\oo)
\ge CK_12^{(R(j)+3r-4)/4\alpha}\e \right\} \tag3.4
\endalign
$$
We shall prove relation (3.1) with the help of (3.4). Let us
first sum for $R(j)=p$ with a fixed $p\ge n_0$. (Observe that the
right-hand side of (3.4) depends on $j$ only through $R(j)$.) We get
that
$$  \allowdisplaybreaks
\align
&\sum_{j\:R(j)=p}\log\frac{B_{m(j)+1}} {B_{m(j-1)+1}}
I\(\frac{\supp_{1\le s\le j}|U_{m(s)}(\oo)|} {A_{m(j)}}>K_1 \e\) \\
&\qquad \le\sum_{j\:R(j)=p} \log\frac{B_{m(j)+1}}{B_{m(j-1)+1}}
\sum_{r=1}^p I\(V_r(\oo) \ge CK_12^{(p+3r-4)/4\alpha}\e\)\\
&\qquad=\sum_{r=1}^p I\(V_r(\oo) \ge CK_12^{(p+3r-4)/4\alpha}\e\)
\sum_{j\:R(j)=p} \log\frac{B_{m(j)+1}} {B_{m(j-1)+1}} \tag3.5\\
&\qquad\le \log\frac{B_{N(p+1)+1}} {B_{N(p)}}
\sum_{r=1}^p I\(V_r(\oo) \ge CK_12^{(p+3r-4)/4\alpha}\e\)\\
&\qquad\le 2\sum_{r=1}^p I\(V_r(\oo) \ge CK_12^{(p+3r-4)/4\alpha}\e\)
\endalign
$$
if $p\ge n_0$ with a sufficiently large threshold $n_0$.
 
We get a good bound on the expression in (3.1) by summing the
estimates (3.5) for $p=L_{n_0}, L_{n_0}+1,\dots$, exploiting that the
terms of (3.1) not considered in such a way give only a bounded
contribution, and the relation $R(j)\le n$ holds for $j\le L_n$. In
such a way we obtain that
$$
\aligned
&\sum_{j=1}^{L_{n}} \log\frac{B_{m(j)+1}} {B_{m(j-1)+1}}
I\(\frac{\supp_{1\le s\le j}|U_{m(s)}(\oo)|}{A_{m(j)}}>K_1 \e\)\\
&\qquad \le 2\sum_{p=1}^n
\sum_{r=1}^p I\(V_r(\oo) \ge CK_1 2^{(p+3r-4)/4\alpha}\e\)+\const
\endaligned \tag3.6
$$
Define the random variables
$$
\chi_r(\oo)=\sum_{p=0}^\infty I\(2^{-r/\alpha}V_r(\oo)
\ge  CK_1 2^{(p-4)/4\alpha}\e \),\quad r=1,2,\dots.
$$
We can write with the help of relation (3.6) by changing the order of
summation at the right-hand side that
$$  \allowdisplaybreaks
\align
&\limsup_{n\to\infty}\frac1n\sum_{j=1}^{L_{n}} \log\frac{B_{m(j)+1}}
{B_{m(j-1)+1}} I\(\frac{\supp_{1\le
s\le j}|U_{m(s)}(\oo)|}{A_{m(j)}}>K_1 \e\)\\
&\qquad\le \limsup_{n\to\infty}\frac2n \sum_{r=1}^n
\sum_{p=r}^n I\(V_r(\oo) \ge CK_12^{r/\alpha} 2^{(p-r-4)/4\alpha}\e\)
\tag3.7 \\
&\qquad=\limsup_{n\to\infty}\frac2n \sum_{r=1}^n
\sum_{p=0}^{n-r} I\(2^{-r/\alpha}V_r(\oo) \ge CK_1
2^{(p-4)/4\alpha}\e\)
\le\limsup_{n\to\infty}\frac2n\sum_{r=1}^n\chi_r(\oo).
\endalign
$$
 
The random variables $\chi_r(\oo)$, $r=1,2,\dots$, are independent, the
relations $0\le E\chi_r(\oo)\le K\e$ and $E\chi_r^2(\oo)\le \const$
hold for $r\ge n_0(\e)$ because of relation (2.5), where an explicit
bound can be given for $K=K(\alpha,\gamma)$. Indeed, the random
variables $\chi_r(\oo)$ take non-negative integer values, the set
$\{\oo\:\chi_r(\oo)\ge k\}$ agrees with the set
$\{\oo\:2^{-r/\alpha} V_r(\oo)\ge CK_12^{(k-4)/4\alpha}\e\}$ whose
probability can be bounded by $C_1\e 2^{-k\gamma/4\alpha}$ by formula
(2.5). (We may assume that $K_1>0$ is chosen so large that
$CK_12^{-1/\alpha}>1$.) This implies that
$P(\chi_r(\oo)\ge x)\le C_1 \e 2^{-\gamma x/4\alpha}$ for $r\ge n_0$.
Hence the laws of large numbers can be applied for these random
variables, and we get that
$$
\align
\lim_{n\to0}&\frac1n\sum_{r=1}^n(\chi_r(\oo)-E\chi_r(\oo))=0,\;
\text{for almost all }\oo\in\Omega \\
&\qquad\text{and}\quad
\limsup_{n\to\infty}\frac1n\sum_{r=1}^n E\chi_r(\oo)\le K\e.
\endalign
$$
These estimates together with relation (3.7) imply (3.1).
 
To prove relation (3.2) let us introduce the random variables
$$
\align
Z^{(i)}_n(\oo)=\frac1{2^{n/\alpha}} \sup_{1\le k\le l_n}
\sup_{1\le p< N(n,k)-N(n,k-1)}&\left|\sum_{m=1}^p
\zeta^{(i)}_{n,k,m}(\oo)\right|,\\
&\qquad n=1,2,\dots,\quad i=1,2.
\endalign
$$
Condition (2.4) implies that
$$
P\(Z^{(i)}_n(\oo)\ge \e x\)\le C_1\e x^{-\gamma}\quad \text{for all
}x\ge A \quad n\ge n_0(\e)\quad \text{and } i=1,2. \tag3.8
$$
We claim that
$$
\align
&\sum_{j=L_{r-1}+1}^{L_{r}}\log\frac
{B_{m(j)+1}}{B_{m(j-1)+1}} I\(\frac{\supp_{1\le
s\le j}\supp_{m(s-1)<p< m(s)}|U_p(\oo)-U_{m(s-1)}(\oo)|}{A_{m(j)}}
>K_2 \e\)\\
&\qquad\qquad\le2 \sum_{u=1}^r\(I\(Z_u^{(1)}(\oo)>
\frac{K_2\e2^{(r-u-1)/\alpha}}2\)+I\(Z_s^{(2)}(\oo)>
\frac{K_2\e2^{(r-u-1)/\alpha}}2\)\)
\endalign
$$
for all $r\ge n_0$. Indeed, the left-hand side of this inequality is
non-zero only if one of the term at the right side is non-zero. In this
case the left hand-side is bounded by
$\summ_{j=L_{r-1}+1}^{L_{r}}\log\dfrac{B_{m(j)+1}}{B_{m(j-1)+1}}\le 2$,
and one of the summands at the right-hand side is non-zero, since
$Z_u^{(1)}(\oo)+Z_u^{(2)}(\oo)> 2^{-u/\alpha}K_2\e A_{m(j)}\ge
K_2\e2^{(r-u-1)/\alpha}$ for some $1\le u\le r$. Hence the inequality
also holds in this case. By summing up this inequality for $r=1,\dots,n$
we get the following bound for the expression in (3.2):
$$
\align
&\limsup_{n\to\infty}\frac1n \sum_{j=1}^{L_{n}}\log\frac
{B_{m(j)+1}}{B_{m(j-1)+1}} I\(\frac{\supp_{1\le
s\le j}\supp_{m(s-1)<p<m(s)}|U_p(\oo)-U_{m(s-1)}(\oo)|}{A_{m(j)}}
>K_2 \e\)\\
&\qquad\le \limsup_{n\to\infty}\frac2n \sum_{r=1}^n\sum_{u=1}^r
\biggl(I\(Z_u^{(1)}(\oo)>\frac{K_2\e2^{(r-u-1)/\alpha}}2\) \tag3.9 \\
&\hskip6.5truecm +I\(Z_u^{(2)}(\oo)>\frac{K_2\e2^{(r-u-1)/\alpha}}2\)
\biggr). \endalign
$$
Let us define the random variables
$$
X_u^{(i)}(\oo)=\sum_{p=0}^\infty I\(Z_u^{(i)}(\oo)\ge\frac{K_2\e}4
2^{(p-1)/\alpha}\), \quad u=0,1,2,\dots,\quad i=1,2.
$$
Then by changing the order of summation at the right-hand side of (3.9)
we get that the left-hand side of formula (3.2) can be bounded by the
expression
$$
\limsup_{n\to\infty}\frac2n\sum_{u=1}^n\(X_u^{(1)}(\oo)+X_u^{(2)}(\oo)\).
$$
The averages of the random variables
$X_u^{(1)}(\oo)+X_u^{(2)}(\oo)-EX_u^{(1)}(\oo)-EX_u^{(2)}(\oo)$ tend to
zero with probability one. Indeed, the random variables $X_u^{(i)}(\oo)$
satisfy the laws of large numbers both for $i=1$ and $i=2$, because
they are independent, and by relation (3.8) the moments of these random
variables are finite. (The estimates $P(X_u^{(i)}>x)\le C_2
2^{-\gamma x/\alpha}$, $i=1,2$, $u\ge n_0$ follows from relation (2.4)
if $K_2>0$ is chosen sufficiently large. This can be proved similarly to
the estimate on the probability of $P(\chi_r(\oo)>x)$ made after
formula (3.7).)  Moreover, $EX_u^{(i)}(\oo)\le K\e$ for all $u\ge
n_0(\e)$ and $i=1,2$, with an appropriate constant $K>0$, and as a
consequence
$$
\limsup_{n\to\infty}\frac2n\sum_{u=1}^n\(EX_u^{(1)}(\oo)+EX_u^{(2)}(\oo)
\)\le 4K\e.
$$
These relations imply formula (3.2). The Basic Lemma is proved.
 
\beginsection 4. The proof of Theorems 1 and 2
 
{\it Proof of Theorem 1.}\/ Let $\eta_n(\oo)$, $n=1,2,\dots$, be a
sequence of independent Gaussian random variables such that
$E\eta_n(\oo)=0$ and $E\eta_n^2(\oo)=\sigma_n^2$. Let us fix a number
$\e>0$. We want to construct a sequence of independent random variables
$\tilde\xi_n^{(\e)}(\oo)$, $n=1,2,\dots$, which has the same
distribution as the sequence $\xi_n(\oo)$, $n=1,2,\dots$, and the
sequences $\zeta_n(\oo)=\tilde\xi^{(\e)}_n(\oo)-\eta_n(\oo)$, and
$U_n(\oo)=\summ_{j=1}^n\zeta_n(\oo)$, $n=1,2,\dots$, satisfy
relation~(2.6) with $B_n=D_n^2=\summ_{k=1}^n\sigma_k^2$, $\alpha=2$ and
the number $\e$ we have fixed. This relation will be proved with an
application of the Basic Lemma. If we can do this for arbitrary $\e>0$,
then Theorem~4 of Part~I., recalled at the beginning of Section~2 and
the almost sure functional (central) limit theorem for the sequence
$\eta_n(\oo)$, $n=1,2,\dots$, imply Theorem~1.
 
We shall omit  the sign ``$\,\tilde{\vphantom{\e}}\,$" and
``${}^{(\e)}$" and write $\xi_n$ instead of $\tilde\xi_n^{(\e)}$. To
apply the Basic Lemma we have to define some quantities. We fix a
sufficiently small $\bar\e=\bar\e(\e)>0$ to be defined later and define
the numbers $N(n)$, $n=1,2,\dots$, by means of the sequence $B_n=D_n^2$
as in the formulation of the Basic Lemma. Then we define an ``$\bar\e$
regular refinement" $N(n,k)$, $n=1,2,\dots$, $0\le k\le l_n$, of the
sequence $N(n)$. By this regularity property we mean that
$$
\aligned
\bar\e (B_{N(n)}-B_{N(n-1)})\le &B_{N(n,k)}-B_{N(n,k-1)}\le 3
\bar\e (B_{N(n)}-B_{N(n-1)})\\
&\qquad \text{for $n\ge n_0(\bar \e)$ and all } 1\le k\le l_n.
\endaligned \tag4.1
$$
The numbers $N(n,k)$ will be defined recursively  in the
variable $k$ for fixed $n$ in the
following way. Put $N(n,0)=N(n-1)$, and if $N(n,k)$ is already defined
and $B_{N(n)}-B_{N(n,k)} >3\bar\e(B_{N(n)}-B_{N(n-1)})$, then
$$
N(n,k+1)=\min\{j\:B_j-B_{N(n,k)}\ge\bar\e (B_{N(n)}-B_{N(n-1)})\}.
$$
If $B_{N(n)}-B_{N(n,k)}\le3\bar\e(B_{N(n)}-B_{N(n-1)})$, then put
$N(n,k+1)=N(n)$. Let us remark that the Lindeberg condition (1.4)
implies that the sequence $B_n$, $n=1,2,\dots$, satisfies relation
(1.1), and $\limm_{N\to\infty}\supp_{N(n-1)\le k\le
N(n)} \dfrac{\sigma_k^2}{B_{N(n)}}=0$. Hence $\limm_{n\to\infty}2^{-n}
B_{N(n)}=1$, and $B_{N(n,k)}-B_{N(n,k-1)}\sim \bar\e
(B_{N(n)}-B_{N(n-1)})$, if $1\le k\le l_n-1$. It is not difficult
to see that the sequence $N(n,k)$ is an $\bar\e$ regular refinement of
the sequence $N(n)$.
 
Let $F_{n,k}(x)=P(S_{n,k}(\oo)<x)$ denote the distribution function
of $S_{n,k}(\oo)=\dfrac1{\bar A_{n,k}}\summ_{j=N(n,k-1)+1}^{N(n,k)}
\xi_j(\oo)$ with $\bar A_{n,k}^2=\summ_{j=N(n,k-1)+1}^{N(n,k)}\sigma_j^2
=B_{N(n,k)}-B_{N(n,k-1)}$, and define the random variables
$T_{n,k}(\oo)=\dfrac1{\bar A_{n,k}}\summ_{j=N(n,k-1)+1}^{N(n,k)}
\eta_j(\oo)$, $n=1,2,\dots$, $k=1,\dots,l_n$. Let $\Phi(x)$ denote the
standard normal distribution function. Then $T_{n,k}(\oo)$,
$n=1,2,\dots$, $k=1,\dots,l_n$, are independent standard normal random
variables, and the variables $\chi_{n,k}(\oo)=\Phi(T_{n,k}(\oo))$ are
independent random variables, uniformly distributed in the interval
$[0,1]$.
 
We shall construct the random variables $S_{n,k}(\oo)=\dfrac1{\bar
A_{n,k}}\summ_{j=N(n,k-1)+1}^{N(n,k)}\xi_j(\oo)$, $n=1,2,\dots$,
$1\le k\le l_n$, by means of the so-called quantile transform as
$S_{n,k}(\oo)=F_{n,k}^{-1}(\chi_{n,k}(\oo))$, where $F_{n,k}^{-1}(x)$
denotes the inverse of the distribution function $F_{n,k}(x)$. More
precisely, we define this inverse function as
$F_{n,k}^{-1}(x)=G_{n,k}(x)=\sup\{u\:F_{n,k}(u)<x\}$, and
$S_{n,k}(\oo)=G_{n,k}(\chi_{n,k}(\oo))$, $n=1,2,\dots$, $k=1,\dots,l_n$.
Such a definition is meaningful for all distribution functions. The
random variables $S_{n,k}(\oo)$, $n=1,2,\dots$, $k=1,\dots,l_n$,
defined in this way are independent, and they have distribution
function $F_{n,k}(x)$.
 
To see that the distribution function of the above defined random
variable $S_{n,k}(\oo)$ is really $F_{n,k}(x)$ let us first observe
that
$$
\align
P(S_{n,k}(\oo)<x)&=\limm_{h\:h>0,h\to0}P(S_{n,k}(\oo)<x-h)\\
&=\limm_{h\:h>0,h\to0}P(G_{n,k}(\chi_{n,k}(\oo))<x-h)\le
P(\chi_{n,k}(\oo)\le F(x))=F(x),
\endalign
$$
since $\{\oo\:G(\chi_{n,k}(\oo))<x-h\}\subset \{\oo\:\chi_{n,k}(\oo)
\le F(x)\}$ for all $h>0$. To see the estimate from the opposite
direction observe that $P(S_{n,k}(\oo)<x)=P(G_{n,k}
(\chi_{n,k}(\oo))<x)\ge P(\chi_{n,k}(\oo)<F(x))=F(x)$, since $\{\oo\:
\chi_{n,k}(\oo)<F(x)\}\subset \{\oo\: G_{n,k}(\chi_{n,k}(\oo))<x\}$.
The last relation holds, since in the case $\chi_{n,k}(\oo)<F_{n,k}(x)$
we have $\chi_{n,k}(\oo)=F_{n,k}(x)-h$ with some $h>0$, and
$G_{n,k}(\chi_{n,k}(\oo))=\supp\{v\: F_{n,k}(v)<F_{n,k}(x)-h\}<x$.
This relation holds, since the continuity of the function
$F_{n,k}(x)$ from the left implies that all numbers $v$ for which
$F_{n,k}(v)<F_{n,k}(x)-h$, the inequality $v\le x-\delta$ holds with
some $\delta=\delta(h)>0$.
 
Define the random variables $S_{N(n,k)}(\oo)=\summ\Sb (m,j)\: m<n-1,\,
1\le j\le l_m\\ \text{or }m=n-1,\text{ and }j\le k\endSb\bar
A_{m,j}S_{m,j}(\oo)$, for all $n=1,2,\dots$, $1\le k\le l_n$. If we
consider the partial sums $S'_n(\oo)=\summ_{k=1}^n\xi_k(\oo)$ with the
random variables $\xi_k(\oo)$ given in the formulation of Theorem~1
then the joint distribution of the random vectors $S_{N(n,k)}(\oo)$ and
$S'_{N(n,k)}(\oo)$, $n=1,2,\dots$, $1\le k\le l_n$ agree. It follows
from the results of general measure theory that in a sufficiently rich
probability space the sequence of random variables of the form
$S_{N(p,k)}(\oo)$, $p=1,2,\dots$, $0\le k\le l_p$, can be extended to a
sequence of random variables $S_n(\oo)$, $n=1,2,\dots$ whose elements
for the indices of the form $N(p,k)$ are the already constructed random
variables $S_{N(p,k)}(\oo)$, and the distribution of the sequences
$S_n(\oo)$ and $S'_n(\oo)$, $n=1,2,\dots$, agree. Define the random
variables $\xi_n(\oo)=S_n(\oo)-S_{n-1}(\oo)$, $n=1,2,\dots$. This
sequence has the same joint distribution as the original sequence of
independent random variables $\xi_n(\oo)$ considered in the formulation
of Theorem~1. In such a way we constructed a sequence of
random variables $\xi_n(\oo)$ in dependence of a small parameter
$\bar\e>0$.
 
Put $\zeta_n(\oo)=\xi_n(\oo)-\eta_n(\oo)$, $n=1,2,\dots$. We claim that
if the parameter $\bar\e=\bar\e(\e)$ is chosen sufficiently small, then
this sequence together with the already constructed ``$\bar\e$
regular refinement" $N(n,k)$ of the sequence $N(n)$ and with the
choice of the random variables $\zeta^{(1)}_n(\oo)=\xi_n(\oo)$,
$\zeta^{(2)}_n(\oo)=\eta_n(\oo)$,
$\zeta^{(1)}_{n,k,m}(\oo)=\xi_{N(n,k-1)+m}(\oo)$ and
$\zeta^{(2)}_{n,k,m}(\oo)=\eta_{N(n,k-1)+m}(\oo)$, $n=1,2,\dots$,
$k=1,\dots,l_n$, $0\le m<N(n,k)-N(n,k-1)$, satisfy the Basic Lemma with
parameter $\e$. This statement implies Theorem~1.
 
The most important step in the proof of this statement is to show the
following estimate. Because of the central limit theorem for all
$\delta>0$ there is a threshold $n_0=n_0(\delta)$ such that
$$
E(S_{n,k}(\oo)-T_{n,k}(\oo))^2\le\delta,\quad \text{for all }n\ge
n_0 \quad\text{and } 1\le k\le l_n. \tag4.2
$$
 
To prove relation (4.2) let us observe that the Lindeberg condition
appearing in the formulation of Theorem~1 makes it possible to apply the
central limit theorem for the normalized sums $S_{n,k}(\oo)$. This
result yields that the distribution functions $F_{n,k}$ of the random
variables $S_{n,k}(\oo)$ satisfy the relation
$\limm_{n\to\infty}\supp_{1\le k\le l_n}\supp_{|x|<\infty}
|F_{n,k}(x)-\Phi(x)|=0$. This implies that the random variables
$T_{n,k}(\oo)$ and the $S_{n,k}(\oo)$ constructed by the above
described quantile transform from the random variables
$\chi_{n,k}(\oo)=\Phi(T_{n,k}(\oo))$ satisfy the following relation:
For all $L>1$ and $\eta>0$ there exists some $n_0=n_0(L,\eta)$ such that
$$
|S_{n,k}(\oo)-T_{n,k}(\oo)|<\frac\eta L\quad \text{on the set }
\{\oo\: |T_{n,k}(\oo)|\le L\}
$$
for all $n\ge n_0$ and $1\le k\le l_n$. Let us choose the number $L>1$
in such a way that the standard normal random variables $T_{n,k}(\oo)$
satisfy the inequality $ET^2_{n,k}(\oo)I(|T_{n,k}(\oo)|\ge
L)<\dfrac\delta{10}$, and let $\eta=\dfrac\delta{20}$. Then
$E(T_{n,k}(\oo)-S_{n,k}(\oo))^2I(|T_{n,k}(\oo)|\le
L)<\dfrac{\delta^2}{400}$, and
$$
\align
&\left|E(T^2_{n,k}(\oo)-S^2_{n,k}(\oo))I(|T_{n,k}(\oo)|\le L)\right|
\le \(E\(T_{n,k}(\oo)+S_{n,k}(\oo)\)^2\)^{1/2}\\
&\qquad\qquad \(E\(T_{n,k}(\oo)-S_{n,k}(\oo)\)^2I(|T_{n,k}(\oo)|\le
L)\)^{1/2} <\dfrac\delta{10}.
\endalign
$$
Since $ES_{n,k}^2(\oo)=ET_{n,k}^2(\oo)=1$, the
inequalities $ES_{n,k}^2(\oo)I(|T_{n,k}(\oo)|\le L)\ge
1-\dfrac\delta5$, and $ES_{n,k}^2(\oo)I(|T_{n,k}(\oo)|>L)\le
\dfrac\delta5$ hold. Because of the simple inequality
$$
\align
E(S_{N(n,k)}(\oo)&-T_{N(n,k)}(\oo))^2\le E(S_{N(n,k)}(\oo)-
T_{N(n,k)}(\oo))^2I(|T_{n,k}(\oo)|\le L)\\
&\quad +2ES_{N(n,k)}^2(\oo)
I(|T_{n,k}(\oo)|\ge L)+2ET_{N(n,k)}^2(\oo))^2 I(|T_{n,k}(\oo)|\ge L)
\endalign
$$
the above relations imply (4.2).
 
Let us remark that the estimate (4.2) does not depend on the parameter
$\bar\e$ appearing in our construction. Only the threshold index
$n_0=n_0(\delta)=n_0(\delta,\bar\e)$ in (4.2) depends on this
parameter.
 
Put
$$
\align
Z_{n,k}(\oo)=\sum_{j=N(n,k-1)+1}^{N(n,k)}\zeta_j(\oo)&=
\sum_{j=N(n,k-1)+1}^{N(n,k)}(\xi_j(\oo)-\eta_j(\oo))\\
&\qquad\qquad\qquad n=1,2,\dots, \; 1\le k\le l_n
\endalign
$$
Observe that $EZ_{n,k}(\oo)=0$, $EZ_{n,k}^2(\oo)\le \delta \bar
A^2_{n,k}$, if $n\ge n_0$, $\summ_{k=1}^{l_n}\bar
A_{n,k}^2=\summ_{j=N(n-1)+1}^{N(n)}\sigma_j^2=B(N(n))-B(N(n-1))
=2^{n-1}(1+o(1))$, and the expression to be estimated in formula (2.5)
can be written in the form $V_n(\oo)=\supp_{1\le j\le
l_n}\left|\summ_{l=1}^j Z_j(\oo)\right|$. These relations together with
the Kolmogorov inequality imply that
$$
P(|V_n(\oo)|\ge \e x2^{n/2})=P\(\supp_{1\le j\le l_n}
\left|\summ_{l=1}^j Z_j(\oo)\right|\ge\e x2^{n/2}\)\le
\frac{\delta\summ_{j=1}^{l_n}\bar A^2_{n,j}}{\e^2x^2 2^n}
\le\frac{2\delta}{\e^2x^2}
$$
Hence we get relation (2.5) with $\gamma=2$ by choosing $\delta=\e^3$.
 
Now we turn to the proof of formula (2.4). We shall prove it only for
the random variables $\zeta^{(1)}_{n,k,p}(\oo)=\xi_{N(n,k-1)+p}(\oo)$,
$n=1,2,\dots$, $k=1,\dots,l_n$, $0\le p< N(n,k)-N(n,k-1)$. The same
proof also applies to the sequence $\zeta^{(2)}_{n,k,p}(\oo)
=\eta_{N(n,k-1)+p}(\oo)$, but this case can be checked simply, because
the random variables $\zeta^{(2)}_{n,k,p}(\oo)$ are Gaussian. The
appropriate choice of the small parameter $\bar\e=\bar\e(\e)>0$ is
important at this step.
 
Let us observe that by relation (4.1) the norming constants $\bar
A_{n,k}$ satisfy the relation $2^{n-1}\bar\e\le \bar
A_{n,k}^2=B_{N(n,k)}-B_{N(n,k-1)}\le2^{n+2}\bar\e$, and $l_n\le
\dfrac2{\bar\e}$ for $n\ge n_0(\bar\e)$. We have to bound the
probability of the event in formula (2.4). Since the random variables
$\zeta^{(1)}_{n,k,p}(\oo)=\xi_{N(n,k-1)+p}(\oo)$, are independent, we
could apply the Kolmogorov inequality to get an estimate for this
expression. But a direct application of this estimate is not good enough
for our purposes, and we have to apply a more refined argument. It is
the factor $l_n^{-1}$ at the right-hand side of (2.4) which makes the
problem harder.
 
First we show that for all $\delta>0$ and $n>n_0(\delta)$ there exists
some $K=K(\delta)$ such that
$$
ES_{n,k}^2(\oo)I(|S_{n,k}(\oo))|>K)\le4\delta. \tag4.3
$$
Relation (4.3) follows from relation (4.2) and the observations that
there exists some $L>0$ such that $ET_{n,k}^2(\oo)I(T_{n,k}(\oo)\ge
L)\le \delta$ and some $K=K(L)>0$ such that $\{\oo\:|S_{n,k}(\oo)|\ge K\}
\subset\{\oo\: |T_{n,k}(\oo)|\ge L\}$ for all sufficiently large $n$
and $1\le k\le l_n$. The latter statement follows from the special
structure of the quantile transform. Then
$ES_{n,k}^2(\oo)I(|S_{n,k}(\oo))|>K)\le
2ET_{n,k}^2(\oo)I(|S_{n,k}(\oo))|>K)+2E(S_{n,k}(\oo)-T_{n,k}(\oo))^2
\le4\delta$, as we claimed.
 
We can write by the Chebishev inequality and formula (4.3) that
$$
\aligned
P&\(\left|\sum_{p=N(n,k-1)+1}^{N(n,k)}
\zeta^{(1)}_{n,k,p}(\oo)\right| >\e x2^{n/2}\)
=P(\bar A_{n,k}|S_{n,k}(\oo)|>\e x2^{n/2})\\
&\qquad \le ES_{n,k}^2(\oo)I(|S_{n,k}(\oo)|\ge K)
\frac {\bar A^2_{n,k}}{\e^2x^22^n} \le
\frac {4\bar A^2_{n,k}\delta}{\e^2x^2 2^n}\\
&\qquad\qquad\qquad\text{for all }x\ge1,\;\;n\ge n_0,\text{ and }1\le k
\le l_n \endaligned  \tag4.4
$$
provided that $\bar\e>0$ is chosen so small that $\bar A_{n,k}K\le
\e 2^{n/2}$, which relation makes it possible to replace the second
moment of $S_{n,k}(\oo)$ by the second moment of the random variable
$S_{n,k}(\oo)I(|S_{n,k}(\oo)|\ge K)$ in  the above estimate. Such a
choice of $\bar\e$ is possible, since $\bar A_{n,k}^2\le \bar\e
2^{n+2}$ if $n\ge n_0$. (We remark that the constants in the
estimations applied to get (4.4) do not depend on the parameter
$\bar\e$. Only the threshold index $n_0$ depends on it.)
Since $l_n\bar A_{n,k}^2\le 2^{n+3}$ the number $\delta>0$ can be chosen
in such a way that $\delta\bar A_{n,k}^2 l_n\le \e^32^n$ for all
sufficiently large $n$. For instance $\delta=\e^3/8$ is a good choice.
With such a choice of $\delta$ we get a weakened form of formula (2.4)
with $\gamma=2$. Here the supremum is dropped, only the last term
$p=N(n,k)-N(n,k-1)$ is taken in the supremum in expression~(2.4).
 
Formula (2.4) in its original form can be proved for instance with the
help of formula (4.4) and the following maximum inequality (see
e.g.~[8], Lemma~3.21 at p.~45). Let $\xi_1(\oo),\dots,\xi_n(\oo)$ be
independent random variables, and put
$S_k(\oo)=\summ_{j=1}^k\xi_j(\oo)$.
Then for all $y>0$ such that $\max\limits_{1\le k\le
n}P\(S_k(\oo)>y\)\le\dfrac14$
$$
P\(\supp_{1\le k\le n}S_k(\oo)\ge 2y\)\le \dfrac43P(S_n(\oo)\ge
y).\tag4.5
$$
We apply this inequality for the partial sums
$$
S_m(\oo)=\pm\summ_{p=1}^m \zeta^{(1)}_{n,k,p}(\oo),\quad 1\le m\le
N(n,k)-N(n,k-1).
$$
Since $E\zeta^{(1)}_{n,k,p}(\oo)=0$, and the variance
$D_n^2=\summ_{p=N(n,k-1)+1}^{N(n,k)-N(n,k-1)}
E(\zeta^{(1)}_{n,k,p})^2(\oo)=\bar A_{n,k}^2$, satisfies the inequality
$D_n^2=\bar A^2_{n,k}\le \bar\e 2^{n+2}$.
Because of these estimates Chebishev's inequality yields that for $n\ge
n_0(\bar\e)$
$$
\max\limits_{1\le m\le N(n,k)-N(n,k-1)}P\(S_m(\oo)>\e \frac x2
2^{n/2}\)\le\frac14 \quad \text{for }x\ge\frac12
$$
if $\bar\e=\bar \e(\e)$ is sufficiently small. Hence formula (4.5) is
applicable with $y=\dfrac x2 2^{n/2}$ if $x\ge\frac12$. Let us observe
that relation (4.4) also holds for $x\ge1/2$ and not only for $x\ge1$.
These relations imply that for sufficiently small $\bar\e=\bar\e(\e)>0$
$$
\align
P&\(\sup_{1\le m\le N(n,k)-N(n,k-1)}\left|\sum_{p=1}^m
\zeta^{(1)}_{n,k,p}(\oo)\right| >\e x2^{n/2}\)\le
\frac {50\bar A^2_{n,k}\delta}{\e^2x^2 2^n}\\
&\qquad\qquad \text{for all }x\ge1,\;\;n\ge0,\text{ and }1\le
k\le l_n.
\endalign
$$
This relation implies formula (2.4) with $\gamma=2$ in the same way as
formula (4.4) implied its weakened form. Theorem 1 is proved.
\medskip\noindent
{\it Proof of Theorem 2.}\/ We may assume that
$\limm_{x\to\infty}\mu(x)=E\xi_1^2(\oo)=\infty$, because in the case
$E\xi_1^2(\oo)<\infty$ Theorem~1 can be applied. More precisely
Theorem~1 supplies a modified version of Theorem~2 in this case with
the norming sequence $\bar B_n=nE\xi_1^2(\oo)$ instead of the original
sequence $B_n$. But $\limm_{n\to\infty}\dfrac{\bar B_n}{B_n}=1$, in
this case, and Theorem~5 of Part~I. implies the result in this case.
 
Since the function $\mu(x)=E\xi_1^2(\oo)I(|\xi_1(\oo)|\le x)$ is a
slowly varying function, it follows for instance from Theorem~2 of [13]
in Chapter~VIII, Section~9 (and its proof) that
$$
\aligned
P(|\xi_1(\oo)|>x)&=o\(x^{-2}\mu(x)\)\quad \text{if }x\to\infty, \\
E|\xi_1|I(|\xi_1(\oo)|>x)&=o\(x^{-1}\mu(x)\)\quad \text{if }x\to\infty.
\endaligned \tag4.6
$$
Define the random variables $\bar\xi_n(\oo)=\xi_n(\oo)I(|\xi_n(\oo)|\le
a_n)$, $\chi_n^{(1)}(\oo)=\xi_n(\oo)I(\xi_n(\oo)\ge a_n)$,  and
$\chi_n^{(2)}(\oo)=\xi_n(\oo)I(\xi_n(\oo)\le-a_n)$, $n=1,2,\dots$, with
the numbers $a_n$ defined in the formulation of ~Theorem~2. We claim
that \medskip
\item{a.)} The sequence of random variables $\bar\xi_n(\oo)$,
$n=1,2,\dots$, satisfies the almost sure functional limit theorem with
the weight function $B_n=\summ_{k=1}^n a_k$, $n=1,2,\dots$, parameter
$\alpha=2$, and the Wiener measure $\mu_0$ as the limit measure.
\item{b.)} Both sequences of random variables $\chi_n^{(i)}(\oo)$,
$n=1,2,\dots$, $i=1,2$, satisfy the almost sure functional limit
theorem with the same weight function $B_n$ as in Statement~a.),
$n=1,2,\dots$, parameter $\alpha=2$, and (degenerated) limit measure
$\mu_0$ on the space $D([0,1])$ which is concentrated on the function
$x(t)\equiv0$.
 
Statement b.) can be reformulated in the following way: Put
$S^{(i)}_n(\oo)=\summ_{k=1}^n \chi_k^{(i)}(\oo)$, $i=1,2$,
$n=1,2,\dots$, and define the broken lines $S_N^{(i)}(\cdot,\oo)$,
$i=1,2$, $N=1,2,\dots$, with the above random variables
$S^{(i)}_n(\oo)$, the numbers $B_n$ and parameter $\alpha=2$ by
formula (1.2). For all $\oo\in\Omega$, $i=1,2$, and $n=1,2,\dots$,
introduce the (random) measure $P_{\oo,i,n}$ by the formula
$P_{\oo,i,n}\(S_k^{(i)}(\cdot,\oo)\)=\dfrac1{\log\frac{B_n}{B_1}}
\log\dfrac{B_{k+1}}{B_k}$, $1\le k<n$. Then for any open neighbourhhood
$\bold G$ of the function $x(t)\equiv0$ in the space $D([0,1])$
$\limm_{n\to\infty}P_{\oo,i,n}(\bold G)=0$ for almost all
$\oo\in\Omega$, $i=1,2$. \medskip
Since $\xi_n(\oo)=\bar\xi_n(\oo)+\chi_n^{(1)}(\oo)+\chi_n^{(2)}(\oo)$,
Statements~a.) and~b.) imply Theorem~2. Indeed, it can be seen for
instance with the help of Lemma~B of Part~I.) and Statement~b.) that for
almost all $\oo\in\Omega$ the sequence $\xi_n(\oo)$, $n=1,2,\dots$,
satisfies the same almost sure functional limit theorem as the sequence
$\bar\xi_n(\oo)$, $n=1,2,\dots$.
 
We shall prove Statement~a.) with the help of Theorem~1. Let us first
observe that $\limm_{n\to\infty}\dfrac{\text{Var}\,\bar\xi_n(\oo)}
{\mu(a_n)}=1$. Indeed, $\dfrac{\text{Var}\,\bar\xi_n(\oo)}{\mu(a_n)}-1
=\dfrac{(E\bar\xi_n(\oo))^2}{\mu(a_n)}$, and by the relations (4.6) and
$E\xi_n(\oo)=0$, the definition of the sequence of $a_n$ and
$a_n\to\infty$ as $n\to\infty$ we have
$(E\bar\xi_n(\oo))^2=(E\xi_n(\oo)I(|\xi_n(\oo)|>a_n)^2
=o(a_n^{-2}\mu(a_n)^2)=o\(\dfrac{\mu(a_n)}n\)$. This means that for
$\bar B_n=\summ_{k=1}^n\text{Var}\,\bar\xi_k(\oo)$,
$\limm_{n\to\infty}\dfrac{\bar B_n}{B_n}=1$.
 
To prove Statement a.) first we show that for all $\e>0$
$$
\lim_{n\to\infty}\frac1{B_n}\sum_{k=1}^n
E\[(\bar\xi_k(\oo)-E\bar\xi_k(\oo))^2I\(|\bar\xi_k(\oo)-E\bar\xi_k(\oo)|
>\e B_n^{1/2}\)\]=0, \tag4.7
$$
which implies that the Lindeberg condition holds for the sequence
$\bar\xi_k(\oo)-E\bar\xi_k(\oo)$, $k=1,2,\dots$.
 
First we show that
$$
\sqrt c\le\liminf_{n\to\infty}\sup_{cn\le k\le n}\frac{a_k}{a_n}\le
\limsup_{n\to\infty}\sup_{cn\le k\le n}\frac{a_k}{a_n}\le1\quad\text{for
all } 0<c<1. \tag4.8
$$
Indeed, it follows from the definition of the sequence $a_n$ that
$a_k\le a_n$ for all $k\le n$, and this implies the right-hand side of
(4.8). On the other hand, since $\mu(\cdot)$ is a slowly varying
function the numbers $a_n$ satisfy the relation $\limm_{n\to\infty}n
\dfrac{\mu(a_n)}{a_n^2}=1$, and for any $\e>0$, $k\ge cn$ and $n\ge
n_0(c,\e)$, $k\dfrac{\mu((\sqrt c-\e)a_n)}
{((\sqrt c-\e)a_n)^2}> cn\dfrac{\mu(a_n)}{(c-\sqrt c\e)a_n^2}\ge1$,
hence $a_k\ge(\sqrt c-\e)a_n$. This relation implies the left-hand side
of (4.8). Since $\mu(\cdot)$ is a slowly varying function, it follows
from relation (4.8) that
$$
\lim_{n\to\infty}\frac{B_n}{n\mu(a_n)}=1. \tag4.9
$$
 
By relation (4.9) the expression $B_n$ can be replaced by $n\mu(a_n)$ in
(4.7). Let us also observe that $B_n^{1/2}\sim\sqrt{n\mu(a_n)}
=\sqrt{\dfrac{n\mu(a_n)}{a_n^2}}a_n\sim a_n$, and $|E\bar\xi_k(\oo)|\le
E|\bar\xi_1(\oo)|\le \const$ for large $n$. Hence the terms in the sum
(4.7) can be estimated as
$$
\align
&E\[(\bar\xi_k(\oo)-E\bar\xi_k(\oo))^2
I\(|\bar\xi_k(\oo)-E\bar\xi_k(\oo)|\ge\e B_n^{1/2}\)\]\\
&\qquad\le2 \(E|\xi_1(\oo)|\)^2+2E \xi_k(\oo)^2I\(\dfrac\e2
a_n\le|\xi_k| \le a_k\) \\
&\qquad\le \const+\max\(\mu(a_k)-\mu\(\frac\e2a_n\),0\).
\endalign
$$
Since $\mu(x)$ is a slowly varying function tending to infinity as
$x\to\infty$, $\limm_{n\to\infty} a_n=\infty$, and $a_k\ge\const\sqrt c
a_n$ if $k\ge cn$, with some $0<c\le 1$ the above estimate implies that
for all $\e>0$ and $\delta>0$ there is some threshold
$n_0=n_0(\e,\delta)$ such that for all $n\ge n_0$ and $1\le k\le n$
$$
E\[(\bar\xi_k(\oo)-E\bar\xi_k(\oo))^
2I\(|\bar\xi_k(\oo)-E\bar\xi_k(\oo)|\ge\e
B_n^{1/2}\)\]\le\delta\mu(a_n).
$$
Summing these inequalities for all $1\le k\le n$ and exploiting that
they hold (for sufficiently large $n$) for all $\delta>0$ we get
relation (4.7).
 
The above relations together with Theorem~5 of Part~I. of this paper
(which states that the weight function $B_n$ in the almost sure
functional limit theorem can be replaced by a weight function $\bar B_n$
such that $\limm_{n\to\infty}\dfrac{\bar B_n}{B_n}=1$) imply that the
sequence of random variables $\bar\xi_n(\oo)-E\bar\xi_n(\oo)$,
$n=1,2,\dots$, satisfy the almost sure functional limit theorem with
parameter $\alpha=2$ and the Wiener measure $\mu_0$ as limit measure.
Hence to finish the proof Statement a.) it is enough to show that
$$
\lim_{n\to\infty}\frac1{B_n^{1/2}}\sum_{k=1}^nE\bar\xi_k(\oo)=0.
\tag4.10
$$
To prove relation (4.10) let us first observe that because of the
identity $E\xi_1(\oo)=0$, relations (4.6) and (4.9) we can write,
fixing an $\e>0$ for all $r>r_0(\e)$ that
$$
\left|\sum_{j=2^r+1}^{2^{r+1}}E\bar\xi_j(\oo)\right|
\le \e\sum_{j=2^r+1}^{2^{r+1}}\frac{\mu(a_j)}{a_j}\le \const\e
2^r\frac{\mu(a_{2^r})}{a_{2^r}}\le \const\e a_{2^r}.
$$
Given an integer $n$, choose the integer $R$ such that $2^R<n\le
2^{R+1}$, and apply the above estimate for all $r_0(\e)\le r\le R$.
Then we get by using the relations $B_n^{1/2}\sim\sqrt{n\mu(a_n)}\sim
a_n$ and $a_n\to\infty$ if $n\to\infty$ that  $\limm_{r\to\infty}
\dfrac{a_{2^{r+1}}}{a_{2^r}}=\sqrt2$ and
$$
\align
\left|\sum_{j=1}^{n}E\bar\xi_j(\oo)\right|&\le
\const(\e)+\const\e\sum_{r=1}^R a_{2^r}\\
&\le\const(\e)+\const'\e a_n\le\const(\e)+\const'\e B_n^{1/2}.
\endalign
$$
Since the above relation holds for all $\e>0$, it implies relation~(4.10).
 
We shall prove Statement~b.) for the sequence $\chi_n^{(1)}(\oo)$,
$n=1,2,\dots$, the proof for the sequence $\chi_n^{(2)}(\oo)$ is the
same. Put $S_n(\oo)=\summ_{j=1}^n\chi^{(1)}_j(\oo)$. First we show
that to verify Statement~b.) it is enough to prove the inequality
$$
\limsup_{n\to\infty}\frac1n\sum_{k=1}^{N(n)}\log\frac
{B_{k}}{B_{k-1}} I\(\left\{\frac{\supp_{0\le
j\le k}S_j(\oo)}{B_k^{1/2}}>\e\right\}\)\le \e, \quad \text{for all }
\e>0 \tag4.11
$$
with $N(n)=\inf\{k\: B_k\ge 2^n\}$. One can argue for instance in the
following way. Let us remark that the sequence of (degenerated)
random variables $\eta_n(\oo)\equiv0$ satisfies the almost sure
functional limit theorem with the weight function $B_n$, parameter
$\alpha=2$ and limit measure $\mu_0$ which is concentrated to the
function $x(t)\equiv0$, $0\le t\le1$. Then to prove Statement~b.) it is
enough to check that the pair of sequences
$(\chi^{(1)}_n(\oo),\eta_n(\oo))$, $n=1,2,\dots$, satisfy Property~A.
Formula (4.11) agrees with Property~A in the present case.
 
We shall prove formula (4.11) by means of the Basic Lemma with the same
sequence of numbers $B_n$ which appears in Theorem~2,
$\zeta_n(\oo)=\chi_n^{(1)}(\oo)$, $n=1,2,\dots$, and the (trivial)
refinement of the sequence $N(n)$ for which the numbers $N(n,k)$
contain all integers in the interval $[N(n-1),N(n)]$. (This refinement
of the sequence $N(n)$ does not depend on $\e$.) With this choice of
the refinement of the sequence $N(n)$ condition (2.4) is an empty
statement in the application of the Basic Lemma, and it is enough to
check relation (2.5). In the present case
$$
V_n(\oo)=\supp_{N(n-1)<k\le N(n)}
\left|\sum_{j=N(n-1)+1}^k\zeta_j(\oo)\right|=
\sum_{j=N(n-1)}^{N(n)}\zeta_j(\oo), \tag4.12
$$
and we have to prove relation (2.5) with this random variable,
$\alpha=2$ and appropriate $\gamma>0$. Then the validity of relation
(2.6) for all $1>\e>0$ implies relation (4.11) i.e.\ the remaining part
of the proof of Theorem~2.
 
We shall prove relation (2.5) with some fixed $\e>0$ and $x\ge1$ for
the sequence $\zeta_n(\oo)=\chi_n^{(1)}(\oo)$ by estimating the
expression $\summ_{j=N(n-1)+1}^{N(n)}E \! \zeta_j(\oo) \! =
E\left|\summ_{j=N(n-1)+1}^{N(n)}\zeta_j(\oo)\right|$.
 
To do this we need a good bound on the term $N(n)$. By the relation
$B_n\sim n\mu(a_n)$ and the definition of the numbers $N(n)$ we have
$N(n)\mu(a_{N(n)})\sim2^{n}$ for sufficiently large $n$.  The relation
$a_{N(n)}^2\dfrac{N(n)\mu(a_{N(n)})} {a_{N(n)}^2}\sim 2^n$ and the
definition of the numbers $a_n$ imply that
$\limm_{n\to\infty}2^{-n/2}a_{N(n)}=1$. Hence
$N(n)\le\dfrac{2^{n+2}}{\mu(a_{N(n)})}$ and
$2^{(n-1)/2}<a_{N(n)}<2^{(n+1)/2}$ for large $n$. Now
we can write with the help of relation (4.6)
$$
\sum_{j=N(n-1)+1}^{N(n)}E\zeta_j(\oo)\le \bar\e
N(n)\frac{\mu(a_{N(n)})}{a_{N(n)}}\le  \bar\e\frac{2^{n+2}}
{a_{N(n)}}\le  \bar\e 2^{(n+5)/2}\le\e^2 2^{n/2}
$$
if the number $\bar\e$ is sufficiently small (and the threshold
$n_0(\bar\e)$ is sufficiently large). Hence the Markov inequality yields
that
$$
P\(\sum_{j=N(n-1)+1}^{N(n)}\zeta_j(\oo)\ge\e x 2^{n/2}\)\le
\dfrac{\e^2 2^{n/2} }{\e x 2^{n/2}}=\e x^{-1}.
$$
It follows from this estimate that the random variables $V_n(\oo)$
defined in (4.12) satisfy the estimate (2.5) with $\gamma=1$. Theorem~2
is proved.
 
\beginsection 5. The proof of Theorem 3
 
{\it Proof of Theorem 3.}\/ By the results quoted in Sections~1 and~2 it
is enough to show that the sequences $(\xi_n(\oo)-a_n,\eta_n(\oo))$,
$n=1,2,\dots$, satisfy Property~A with an appropriate sequence of
constants~$a_n$ and $\eta_n(\oo)=\bar L(n)^{1/\alpha}\bar\eta_n(\oo)$,
$n=1,2,\dots$, where $\bar\eta_n(\oo)$ are i.i.d.\ random variables
with the stable distribution $G(x)$ satisfying formula~(1.5), and $\bar
L(\cdot)$ is the slowly varying function defined in the formulation of
Theorem~3 whose existence still has to be proved.
 
We shall prove Property~A with the help of the Basic Lemma together with
a quantile transform representation of the random variables
$\xi_n(\oo)$, by means of the random variables $\eta_n(\oo)$,
$n=1,2,\dots$, to be described below.
 
The random variables $\eta_n(\oo)$, $n=1,2,\dots$, have distribution
function $G_n(x)=G\(\dfrac x{\bar L(n)^{1/\alpha}}\)$ with the stable
distribution function $G(x)$ which satisfies relation (1.5). The
distribution function $G_n(x)$ has a density function, hence the
independent random variables $G_n(\eta_n(\oo))$, $n=1,2,\dots$, are
uniformly distributed in the interval $[0,1]$. Then, similarly to the
construction in the proof of Theorem~1 we can construct the random
variables $\xi_n(\oo)$, $n=1,2,\dots$. More explicitly, we define a
sequence of i.i.d.\ random variables with the same distribution
function $F(x)$ as the originally given sequence $\xi_n(\oo)$,
$n=1,2,\dots$, by the formula $\xi_n(\oo) =F^{-1}(G_n(\eta(\oo))$,
$n=1,2,\dots$, where $F^{-1}(x)=\sup\{u\:F(u)<x\}$. We will show
with the help of the Basic Lemma that this construction of the pairs
$(\xi_n(\oo)-a_n,\eta_n(\oo))$, $n=1,2,\dots$, with
$$
a_n=E\(\xi_n(\oo)-\eta_n(\oo)\)I\(|\eta_n(\oo)|<n^{1/\alpha}
\bar L(n)^{1/\alpha}\)
$$
satisfies Property~A. (Observe that the construction which must satisfy
certain property depending on a parameter $\e>0$ does not depend on the
parameter~$\e$.)
 
First we formulate and prove a Lemma which plays an important role in
the proof.\medskip\noindent
{\bf Lemma 1.} {\it Let us consider the function
$b(x)=\max\left\{u\:\dfrac{L(u)x}{u^\alpha}\ge1\right\}$,
$x>0$, introduced in the formulation of Theorem~3. This function $b(x)$
is a regularly varying function with parameter $1/\alpha$, hence
$\bar L(x)=b(x)^{\alpha}x^{-1}$ is a slowly varying function.
Define the distribution functions $G_n(x)$ and random variables
$\eta_n(\oo)$ and $\xi_n(\oo)$, $n=1,2,\dots$, in the way described
above  with the help of this slowly varying function
$\bar L(\cdot)$. The relation
$\limm_{x\to\infty}\dfrac{xL(b(x))}{b(x)^{\alpha}}=1$
holds. For all $\bar\e>0$ there exists an index $n_0=n_0(\bar\e)$ such
that
$$
\aligned
|\xi_n(\oo)-\eta_n(\oo)|<\bar\e|\eta_n(\oo)|\quad&\text{if } \bar\e
n^{1/\alpha} \bar L(n)^{1/\alpha}<\eta_n(\oo)<\bar\e^{-1}
n^{1/\alpha} \bar L(n)^{1/\alpha}\\
&\qquad\text{and }n\ge n_0(\bar\e).
\endaligned \tag5.1
$$}
\noindent
{\it Proof of Lemma 1.}\/ Since $L(x)$ is a slowly varying function we
have $\limm_{x\to\infty}b(x)=\infty$, and the
relations $\limm_{x\to\infty}\dfrac{L(b(x))x}{b(x)^\alpha}=1$ and
$\limm_{x\to\infty} \dfrac{L(c^{1/\alpha}b(x))cx}{(c^{1/\alpha}
b(x))^{\alpha}}=1$ hold for all $1\le c\le K$, where $K>1$ is an
arbitrary fixed constant. Moreover, the convergence in the second
relation is uniform in the variable $c$ as $x\to\infty$. It can be
shown with the help of these properties that
$\limm_{x\to\infty}\dfrac{b(cx)}{c^{1/\alpha}b(x)}=1$, since they imply
that for large $x>0$ the number $c^{1/\alpha}b(x)$ is a good
approximation for $b(cx)$.  This relation means that the function $b(x)$
is regularly varying with parameter $1/\alpha$.
The regular varying property of the function $b(x)$ implies that the
function $\bar L(x)=b(x)^\alpha x^{-1}$ is slowly varying.
 
We claim that for any $\bar\e>0$
$$
\aligned
&\lim_{n\to\infty}\sup_{\frac{\bar\e}2 n^{1/\alpha}
\bar L(n)^{1/\alpha}<x<2\bar\e^{-1}n^{1/\alpha}\bar
L(n)^{1/\alpha}}x^\alpha\dfrac{1-F(x)}{L(x)}=C_1 \\
&\lim_{n\to\infty}\sup_{\frac{\bar\e}2 n^{1/\alpha}\bar
L(n)^{1/\alpha}<x<2\bar\e^{-1}
n^{1/\alpha}\bar L(n)^{1/\alpha}}x^\alpha\dfrac{F(-x)}{L(x)}=C_2,
\endaligned \tag5.2
$$
and
$$
\aligned
&\lim_{n\to\infty}\sup_{\frac{\bar\e}2 n^{1/\alpha}\bar
L(n)^{1/\alpha}<x<2\bar\e^{-1} n^{1/\alpha}\bar
L(n)^{1/\alpha}}x^\alpha\dfrac{1-G_n(x)}{L(x)}=C_1 \\
&\lim_{n\to\infty}\sup_{\frac{\bar\e}2 n^{1/\alpha}\bar
L(n)^{1/\alpha}<x<2\bar\e^{-1}
n^{1/\alpha}\bar L(n)^{1/\alpha}}x^\alpha\dfrac{G_n(-x)}{L(x)}=C_2.
\endaligned\tag$5.2'$
$$
Formula (5.2) follows from relation (1.6). Formula
$(5.2')$ can be deduced from (1.5) and the definition of the function
$G_n(x)$ if we show that
$$
\limm_{n\to\infty} \supp_{\frac{\bar\e}2 n^{1/\alpha}\bar
L(n)^{1/\alpha}<x <2\bar\e^{-1} n^{1/\alpha}\bar
L(n)^{1/\alpha}}\dfrac{\bar L(n)}{L(x)}=1.
$$
To see the last statement observe that in the domain we are interested
in $L(x)\sim L(n^{1/\alpha}\bar L(n)^{1/\alpha}))$, hence it can be
reduced to the formula
$$
\limm_{n\to\infty}\dfrac{\bar
L(n)}{L(n^{1/\alpha}\bar L(n)^{1/\alpha})}=1. \tag 5.3
$$
Relation (5.3) holds, since $L(n^{1/\alpha}\bar L(n)^{1/\alpha})=
L(b(n))\sim\dfrac{b(n)^{\alpha}}n=\bar L(n)$.
 
Relations (5.2) and $(5.2')$ imply that for all $\bar\e>0$ there is a
threshold $n=n(\bar\e)$ such that
$$
\align
1-F((1+\bar\e)x)&<1-G_n(x)<1-F((1-\bar\e)x)\\
F(-(1+\bar\e)x)&<G_n(-x)\hphantom{1\,\,}<F((1-\bar\e)x)\\
&\qquad \text{if } \bar\e n^{1/\alpha} L(n)^{1/\alpha}<x<
\bar\e^{-1} n^{1/\alpha} L(n)^{1/\alpha}.
\endalign
$$
Hence
$$
\align
(1-\bar\e)x&<F^{-1}(G_n(x))<(1+\bar\e)x\\
-(1+\bar\e)x&<F^{-1}(G_n(-x))<-(1-\bar\e)x
\endalign
$$
if $\bar\e n^{1/\alpha} L(n)^{1/\alpha}<x< \bar\e^{-1} n^{1/\alpha}
L(n)^{1/\alpha}$. The last formula together with the definition of the
quantile transform imply relation~(5.1). Lemma~1 is proved. \medskip
 
Now we turn back to the proof of Theorem~3. We shall prove Property~A
with the help of the Basic Lemma with the (greatest possible) refinement
$N(n,k)$ of the sequence $N(n)$, $n=1,2,\dots$, for which the numbers
$N(n,k)$ contain all integers in the interval $[N(n-1),N(n)]$ for all
$n=1,2,\dots$. With this choice of the sequence $N(n,k)$ condition (2.4)
has not be checked in the application of the Basic Lemma. We only have
to check formula (2.5), which states in the present case that
$$
\aligned
P&\(\sup_{N(n-1)<k\le N(n)}\left|\sum_{j=N(n-1)+1}^k
(\xi_j(\oo)-a_j-\eta_j(\oo))\right|>\e x2^{n/\alpha}\)<C\e x^{-\gamma}\\
&\hskip 6truecm \text{for all } x\ge1 \text { and }n\ge n_0(\e)
\endaligned\tag5.4
$$
with an appropriate $\gamma>0$.
 
Let us make the following observation. Since $\bar L(n)$ is a slowly
varying function the relation $\limm_{n\to\infty}\dfrac{\bar
B_n}{B_n}=1$ holds with $\bar B_n=n\bar L(n)$. This relation together
with formula (1.1) imply that for all large $n$ the number $N(n)$
satisfies the relation $N(n)\bar L(N(n))\sim 2^{n}$. Since $\bar
L(n)$ is a slowly varying function this relation also implies that
$N(n)\le 5 N(n-1)$ if $n$ is sufficiently large. Indeed, since
$\dfrac{\bar L(N(n-1))}{\bar L(N(n))}\le1.01\(\dfrac{N(n)}
{N(n-1)}\)^{1/2}$ for sufficiently large $n$, we have $\sqrt
5\ge\dfrac{1.01N(n)\bar L(N(n))} {N(n-1)\bar L(N(n-1))}\ge
\(\dfrac{N(n)}{N(n-1)}\)^{1/2}$ for sufficiently large $n$, as we
claimed. Hence $\bar L(k)\sim\bar L(N(n))$ and $(k\bar L(k))^{1/\alpha}
\le \dfrac98 2^{n/\alpha}$ for $N(n-1)\le k\le N(n)$.
 
In the proof of the inequality (5.4) we shall bound separately the
contribution of those indices $j$ for which $\eta_j(\oo)$ is relatively
small, more explicitly their values is much less than $j^{1/\alpha}\bar
L(j)^{1/\alpha}$, the indices $j$ of middle order terms for which
$\eta_j(\oo)$ is of order $j^{1/\alpha}\bar L(j)^{1/\alpha}$ and finally
those indices for which the terms $\eta_j(\oo)$ are much larger
than $j^{1/\alpha}\bar L(j)^{1/\alpha}$. The contribution of the terms
with small and middle order will be bounded by means of an estimate for
the expected values and variances of their sum together with an
application of Kolmogorov's inequality. The contribution of the large
values can be estimated by means of the tail behaviour of the
distribution functions $F(x)$ and $G_n(x)$. The separation between the
large and middle values will depend also on the value of the
parameter~$x$ in formula~(5.4). In the estimates we shall fix a
sufficiently small $\bar\e=\bar\e(\e)$ and apply the estimate~(5.1).
 
To estimate the contribution of the small terms let us first bound the
terms
$$
E(\xi_j(\oo)-\eta_j(\oo))I\(|\eta_j(\oo)|<\e^{4/(2-\alpha)}
j^{1/\alpha}L(j)^{1/\alpha}\)-a_j
$$
and
$$E(\xi_j(\oo)-\eta_j(\oo))^2
I\(|\eta_j(\oo)|<\e^{4/(2-\alpha)} j^{1/\alpha}\bar L(j)^{1/\alpha}\).
$$
Fix a sufficiently small $\bar\e=\bar\e(\e)$ to be defined later, and
consider indices $j>n_0(\bar\e)$. i.e.\ such indices for which the
relation~(5.1) is applicable. In this case, if $\bar\e(\e)$ is
sufficiently small, we can write because of the definition of the
norming constants $a_n$
$$
\aligned
&\left|E(\xi_j(\oo)-\eta_j(\oo))I\(|\eta_j(\oo)|<\e^{4/(2-\alpha)}
j^{1/\alpha} \bar L(j)^{1/\alpha}\)-a_j\right|\\
&\qquad =\left|E(\xi_j(\oo)-\eta_j(\oo))I\(\e^{4/(2-\alpha)}
j^{1/\alpha} \bar L(j)^{1/\alpha}\le|\eta_j(\oo)|<
j^{1/\alpha} \bar L(j)^{1/\alpha}\)\right|\\
&\qquad \le\bar\e E|\eta_j(\oo)|I\(\e^{4/(2-\alpha)}
j^{1/\alpha} L(j)^{1/\alpha}\le|\eta_j(\oo)|<
j^{1/\alpha} \bar L(j)^{1/\alpha}\)\\
&\qquad=\bar\e\int_{\e^{4/(2-\alpha)}j^{1/\alpha}\bar
L(j)^{1/\alpha}\le
|u|\le j^{1/\alpha}\bar L(j)^{1/\alpha}}|u|G_j(\,du)\\
&\qquad\le\const\bar\e \bar L(j)^{1/\alpha}
\int_{\e^{4/(2-\alpha)}j^{1/\alpha}}^{j^{1/\alpha}}u\bar G(\,du)
\le \frac\e{45} j^{(1-\alpha)/\alpha}\bar L(j)^{1/\alpha}
\endaligned\tag5.5
$$
with $\bar G(u)=1-G(u)+G(-u)$. To estimate the second moment of these
summands write
$$
\align
&E(\xi_j(\oo)-\eta_j(\oo))^2 I\(|\eta_j(\oo)|<\e^{4/(2-\alpha)}
j^{1/\alpha}\bar L(j)^{1/\alpha}\)\\
&\qquad \le 2E\xi_j^2(\oo)I\(|\xi_j(\oo)|<2\e^{4/(2-\alpha)}
j^{1/\alpha}\bar L(j)^{1/\alpha}\)\\
&\qquad\qquad +2E\eta_j^2(\oo)I\(|\eta_j(\oo)|<\e^{4/(2-\alpha)}
j^{1/\alpha}\bar L(j)^{1/\alpha}\).
\endalign
$$
This inequality implies together with an asymptotic relation about
the behaviour of the moments of a random variable with regularly
varying distribution function, a result which follows for instance
from Theorem~2, Part~(i)  in [13] Chapter~VIII. Section~9 with the
choice $\eta=0$ and $\zeta=2$, that
$$
\aligned
&E(\xi_j(\oo)-\eta_j(\oo))^2 I\(|\eta_j(\oo)|<\e^{4/(2-\alpha)}
j^{1/\alpha}\bar L(j)^{1/\alpha}\)\\
&\qquad\le\const \e^{8/(2-\alpha)}j^{2/\alpha}\bar L(j)^{2/\alpha}
\biggl[\(1-F(\e^{4/(2-\alpha)} j^{1/\alpha}\bar L(j)^{1/\alpha})\)\\
&\qquad\qquad\hskip5.5truecm+\(1-G_j(\e^{4/(2-\alpha)} j^{1/\alpha}\bar
L(j)^{1/\alpha})\)\biggr]\\
&\qquad\le\const \e^{8/(2-\alpha)}j^{2/\alpha}\bar L(j)^{2/\alpha}
\e^{-4\alpha/(2-\alpha)}j^{-1}\bar L(j)^{-1}\[L\(j^{1/\alpha}\bar
L(j)^{1/\alpha}\)+\bar L(j)\] \\
&\qquad\le\const\e^4j^{-1+2/\alpha}\bar L(j)^{2/\alpha}
\endaligned\tag5.6
$$
because of formula (5.3).
 
The estimates (5.5) and (5.6) together with the estimates
$N(n)\bar L(N(n))\sim2^n$, $(k\bar L(k))^{1/\alpha}\le \frac98
2^{n/\alpha}$ for $k\le N(n)$, $N(n)<5N(n-1)$ and Kolmogorov's
inequality imply that with the notation of the events
$A_j=\{\oo\:|\eta_j(\oo)|<
\e^{4/(2-\alpha)}j^{1/\alpha}\bar L(j)^{1/\alpha}
\}$ and their indicator functions
$I(A_j(\oo))$, $j=1,2,\dots$,
$$   \allowdisplaybreaks
\align
&\left|\sum_{j=N(n-1)+1}^k
E(\xi_j(\oo)-\eta_j(\oo))I(A_j(\oo))-a_j)\right|
\le \sum_{j=N(n-1)+1}^k\frac\e{45}j^{(1-\alpha)/\alpha}\bar
L(j)^{1/\alpha}\\
&\qquad\le \frac\e{40} 2^{n/\alpha}\sum_{j=N(n-1)+1}^{N(n)}\frac1j
\le\frac\e{30}2^{n/\alpha}\log\frac{N(n)}{N(n-1)}\le\frac\e82^{n/\alpha}
\endalign
$$
for all $N(n-1)<k\le N(n)$, and
$$
\aligned
&P\(\sup_{N(n-1)<k\le N(n)}\left|\sum_{j=N(n-1)+1}^k
(\xi_j(\oo)-\eta_j(\oo))I(A_j(\oo))-a_j\right|>\frac\e4
x2^{n/\alpha}\)\\
&\qquad\le\frac{\summ_{j=N(n-1)+1}^{N(n)}E(\xi_j(\oo)-\eta_j(\oo))^2
I(A_j(\oo))}{\frac{\e^2}{64}x^22^{2n/\alpha}}\le
\frac{\e^4\[N(n)\bar
L(N(n))\]^{2/\alpha}}{\frac{\e^2}{64}x^22^{2n/\alpha}}\\
&\qquad\le\frac{\const\e^2}{x^2} \quad \text{for all } x\ge1
\text { and } n\ge n_0(\bar\e)
\endaligned\tag5.7
$$
 
We still have to bound the contribution of those terms
$\xi_j(\oo)-\eta_j(\oo)$ in the sum (5.4) for which $\eta_j(\oo)\ge
\e^{4/(2-\alpha)}j^{1/\alpha}\bar L(j)^{1/\alpha}$. We shall separate
them to middle terms for which $|\eta_j(\oo)|$ is not too large
and large terms for which $|\eta_j(\oo)|$ is large. This separation to
the middle and large terms will be made differently for large and small
values of the parameter $x$ in formula (5.4).
 
Let us choose the number $\bar\e(\e)$ sufficiently small, and define
the sets
$$
\align
B_j(x)&=\left\{\oo\: \e^{4/(2-\alpha)} j^{1/\alpha}\bar
L(j)^{1/\alpha}\le|\eta_j(\oo)|<\e^{-1/\alpha} x j^{1/\alpha}\bar
L(j)^{1/\alpha}\right\}\\ \intertext{and}
C_j(x)&=\left\{\oo\:|\eta_j(\oo)|\ge\e^{-1/\alpha}x j^{1/\alpha}
\bar L(j)^{1/\alpha}\right\}
\endalign
$$
together with the indicator functions $I(B_j(x)(\oo))$ and
$I(C_j(x)(\oo))$, $j=1,2,\dots$, of these sets if $1\le x\le
\bar\e^{-1/4}$.  For sufficiently large $j$ relation (5.1) can be
applied on the sets $B_j(\oo)$. This implies that because of the factor
$\bar\e$ in the upper bound given (5.1) we can prove similarly to the
proof of relations (5.5) and (5.6) that
$$
\left|E(\xi_j(\oo)-\eta_j(\oo))I(B_j(x)(\oo))\right|
\le \frac\e{45} j^{(1-\alpha)/\alpha}\bar L(j)^{1/\alpha}.
\tag5.8
$$
and
$$
E(\xi_j(\oo)-\eta_j(\oo))^2 I(B_j(x)(\oo))\le
\const\e^4j^{-1+2/\alpha}\bar L(j)^{2/\alpha}\tag5.9
$$
for $x\le \bar\e^{-1/4}$. These estimates together with  Kolmogorov's
inequality and the asymptotic relations we have for $N(n)$ and
$L(N(n)$ yield similarly to the estimate (5.7) the bound
$$
\align
\left|\sum_{j=N(n-1)+1}^k E(\xi_j(\oo)-\eta_j(\oo))I(B_j(\oo))\right|
&\le \sum_{j=N(n-1)+1}^k\frac\e{45}j^{(1-\alpha)/\alpha}\bar
L(j)^{1/\alpha}\\
&\le\frac\e9\(N(n)\bar L(N(n))\)^{1/\alpha}\le\frac\e8 2^{n/\alpha}
\endalign
$$
for all  $N(n-)+1\le k\le N(n)$ and
$$
\aligned
&P\(\sup_{N(n-1)<k\le N(n)}\left|\sum_{j=N(n-1)+1}^k
(\xi_j(\oo)-\eta_j(\oo))I(B_j(x)(\oo))\right|>\frac\e4
x2^{n/\alpha}\)\\
&\qquad\le\frac{\summ_{j=N(n-1)+1}^{N(n)}E(\xi_j(\oo)-\eta_j(\oo))^2
I(B_j(x)(\oo))}{\frac{\e^2}{64}x^22^{2n/\alpha}}\le\const
\frac{\e^4\[N(n)\bar
L(N(n))\]^{2/\alpha}}{\frac{\e^2}{64}x^22^{2n/\alpha}}\\
&\qquad\le\frac{\const\e^2}{x^2} \quad \text{for all }\bar\e^{-1/4}\ge
x\ge1 \text { and } n\ge n_0(\bar\e).
\endaligned\tag5.10
$$
In the outer domain we can write for $\bar\e^{-1/4}\ge x$ because of the
definition of the sets $C_j(x)$
$$
\aligned
&P\(\sup_{N(n-1)<k\le N(n)}\left|\sum_{j=N(n-1)+1}^k
(\xi_j(\oo)-\eta_j(\oo))I(C_j(x)(\oo))\right|>\frac\e2 x2^{n/\alpha}\)\\
&\qquad \le P\(\left\{\oo\: \oo\in C_j(x) \quad \text{for some
}N(n-1)<k\le N(n)\right\}\)\\
&\qquad \le  \summ_{j=N(n-1)+1}^{N(n)}
P\(|\eta_j(\oo)| \ge\e^{-1/\alpha} x j^{1/\alpha}
\bar L(j)^{1/\alpha}\)\\
&\qquad \le\const\e x^{-\alpha} \sum_{j=N(n-1)+1}^{N(n)}\frac1j\le\const
\e x^{-\alpha}. \endaligned\tag5.11
$$
 
Since $A_j\cup B_j(x)\cup C_j(x)=\Omega$ for all $j$, relations (5.7),
(5.10) and (5.11) imply relation (5.4) with $\gamma=\alpha$ in the case
$x<\bar\e^{-1/4}$.
 
In the case $x\ge\bar\e^{-1/4}$ define the sets
$$
\align
B_j'(x)&=\left\{\oo\: \e^{4/(2-\alpha)} j^{1/\alpha}\bar
L(j)^{1/\alpha}\le|\eta_j(\oo)|< x^{1/2} j^{1/\alpha}\bar
L(j)^{1/\alpha}\right\}\\ \intertext{and}
C_j'(x)&=\left\{\oo\:|\eta_j(\oo)|\ge x^{1/2} j^{1/\alpha}\bar
L(j)^{1/\alpha}\right\} \endalign
$$
together with the indicator functions of these sets $I(B'_j(x)(\oo))$
and $I(C'_j(x)(\oo))$, $j=1,2,\dots$. We remark that if $\bar\e(\e)>0$
is very small, then $x^{1/2}\ll \e x$. In this case we can apply the
estimation
$$
L\(x^{1/2} j^{1/\alpha}\bar L(j)^{1/\alpha}\)\le C(t)
x^{t}L(j^{1/\alpha}\bar L(j)^{1/\alpha})\le C'(t)
x^{t}\bar L(j)           \tag5.12
$$
for all $x\ge1$ and $t>0$ which holds, because $L(\cdot)$ is a slowly
varying function. With the help of relation (5.12) the following
(weaker) version of the estimate (5.11) can be proved which is
appropriate for our purposes.
$$
\aligned
&P\(\sup_{N(n-1)<k\le N(n)}\left|\sum_{j=N(n-1)+1}^k
(\xi_j(\oo)-\eta_j(\oo))I(C_j'(x)(\oo))\right|>\frac\e2
x2^{n/\alpha}\)\\
&\qquad\le\summ_{j=N(n-1)+1}^{N(n)+1} P\(|\eta_j(\oo)|\ge
x^{1/2} j^{1/\alpha}\bar L(j)^{1/\alpha}\) \\
&\qquad \le\const x^{-\alpha/3}\sum_{j=N(n-1)+1}^{N(n)}\frac1j\le\const
x^{-\alpha/3}<\e x^{-\alpha/4}, \endaligned \tag5.13
$$
if $\bar\e$ is sufficiently small, and as a consequence $x$ is
sufficiently large.
 
To prove an estimate analogous to (5.10), to bound the  contribution of
the  middle terms for large $x$, we give a (weaker) estimate on the
first two moments of the random variables
$(\xi_j(\oo)-\eta_j(\oo))I(B_j'(x))(\oo)$.
 
The estimation of the second moment is simpler. In this case we can
argue similarly to the estimate (5.6). We can apply the result from
Feller's book [13], Theorem~2, Part~(i) Chapter~VIII.\ Section~9 with
the choice $\eta=0$ and $\zeta=2$. Then we get with the application of
formula (5.12) that
$$
\aligned
&E(\xi_j(\oo)-\eta_j(\oo))^2 I(B_j'(x)(\oo)) \le\const
xj^{2/\alpha}\bar L(j)^{2/\alpha}\\
&\qquad\qquad\biggl[\(1-F(x^{1/2} j^{1/\alpha}\bar L(j)^{1/\alpha})\)
+\(1-G_j(x^{1/2} j^{1/\alpha}\bar L(j)^{1/\alpha})\)\biggr]\\
&\qquad\le\const xj^{2/\alpha}\bar L(j)^{2/\alpha}
x^{-\alpha/2}j^{-1}\bar L(j)^{-1}\[L\(
x^{1/2}j^{1/\alpha}\bar L(j)^{1/\alpha}\)+\bar L(j)\] \\
&\qquad\le\const xj^{-1+2/\alpha}\bar L(j)^{2/\alpha},
\endaligned
$$
and
$$
\text{Var\,}\(\sum_{j=N(n-1)+1}^{N(n)}(\xi_j(\oo)-\eta_j(\oo))
I(B_j'(x))(\oo)\) \le\const x 2^{2n/\alpha}. \tag5.14
$$
To get an appropriate estimate on the first moment let us observe that
in the estimation in formula (5.5) we gave a good bound on the
integral we want to estimate if the domain of integration is restricted
to the interval $\e^{4/(2-\alpha)}{j^{1/\alpha}\bar L(j)^{1/\alpha}\le
|u|\le j^{1/\alpha}\bar L(j)^{1/\alpha}}$. Hence, handling this part of
the integral separately we get integrating by parts and applying (5.12)
with $t=\alpha>0$
$$
\aligned
&\left|E(\xi_j(\oo)-\eta_j(\oo))I(B_j'(x)(\oo))\right|\le
\e j^{-1+1/\alpha}\bar L(j)^{1/\alpha}\\
&\qquad\qquad+\int_{j^{1/\alpha}\bar L(j)^{1/\alpha}\le |u|\le
x^{1/2} j^{1/\alpha}\bar L(j)^{1/\alpha}} |u|(F(\,du)+G_j(du))\\
&\qquad\le\const\int_{j^{1/\alpha}\bar L(j)^{1/\alpha}\le
|u|\le x^{1/2} j^{1/\alpha}\bar L(j)^{1/\alpha}}
(F(u)+G_j(u))\,du\\
&\qquad\qquad+\const x^{1/2} j^{-1+1/\alpha}\bar L(j)^{1/\alpha}
\le\const x^{1/2} j^{-1+1/\alpha}\bar L(j)^{1/\alpha}.
\endaligned
$$
Hence
$$
\aligned
&\left|E\sum_{j=N(n-1)+1}^{k}(\xi_j(\oo)-\eta_j(\oo))
I(B_j'(x)(\oo))\right|\\
&\qquad\le\const x^{1/2} N(n)^{1/\alpha}\bar
L(N(n))^{1/\alpha}\le \const x^{1/2}2^{n/\alpha}
\le\frac\e8  x2^{n/\alpha}
\endaligned \tag5.15
$$
for all $N(n-1)\le k\le N(n)$ if $\bar\e=\bar\e(\e)$ is sufficiently
small, hence $x>\bar\e^{-1/4}$
is sufficiently large. The estimates (5.14) and (5.15) together with
Kolmogorov's inequality imply that
$$
\aligned
&P\(\sup_{N(n-1)<k\le N(n)}\left|\sum_{j=N(n-1)+1}^k
(\xi_j(\oo)-\eta_j(\oo))I(B_j'(x)(\oo))\right|>\frac\e4
x2^{n/\alpha}\)\\
&\qquad\le\frac {x 2^{2n/\alpha}}{\frac{\e^2}{64}x^22^{2n/\alpha}}
\le\frac \e{x^{1/2}} \quad \text{for all }x\ge\bar\e^{-1/4}
\text { and } n\ge n_0(\bar\e)
\endaligned\tag5.16
$$
if $\bar\e$ is sufficiently small. Now relations (5.7), (5.13) and
(5.16) imply relation (5.4) in remaining case $x>\bar\e^{-1/4}$
with $\gamma=\min\(\frac\alpha4,\frac12\)=\frac\alpha4$. Since the
number $\bar\e=\bar\e(\e)$ can be chosen in such a way that all
inequalities needed in the proof are satisfied, the above calculations
imply the almost sure functional limit theorem.
 
To complete the proof of Theorem 3 we still have to show that the
random variables $\dfrac1{B_n}\summ_{j=1}^n(\xi_j(\oo)-a_n)$ converge
in distribution to the distribution function $G(x)$. We prove this if
we show that $\dfrac1{n^{1/\alpha}\bar
L(n)^{1/\alpha}}\summ_{j=1}^n(\xi_j(\oo)-\eta_j (\oo)-a_n)
\Rightarrow0$ as $n\to\infty$, where $\Rightarrow$ denotes
convergence in distribution. (The denominator $B_n$ can be replaced by
$n^{1/\alpha}\bar L(n)^{1/\alpha}\sim B_n$ in this formula.) Let us
choose the representation
$$
\aligned
&\xi_j(\oo)-\eta_j(\oo)=
(\xi_j(\oo))-\eta_j(\oo))I\(|\eta_j(\oo)|<\e^{4/(2-\alpha)}
n^{1/\alpha}\bar L(n)^{1/\alpha}\)\\
&\qquad\qquad+(\xi_j(\oo))-\eta_j(\oo))I\(
\e^{4/(2-\alpha)}n^{1/\alpha}\bar L(n)^{1/\alpha})\le
|\eta_j(\oo)|\le \e^{-1}n^{1/\alpha}\bar L(n)^{1/\alpha}\)\\
&\qquad\qquad+ (\xi_j(\oo))-\eta_j(\oo))I\(|\eta_j(\oo)|\ge\e^{-1}
n^{1/\alpha}\bar L(n)^{1/\alpha}\),
\endaligned \tag5.17
$$
and apply a natural modification of relations (5.5), (5.6), (5.8) and
(5.9) appropriate in the present case. Here
the separation levels are chosen as $\e^{4/(2-\alpha)}n^{1/\alpha}\bar
L(n)^{1/\alpha}$ instead of $\e^{4/(2-\alpha)}j^{1/\alpha}\bar
L(j)^{1/\alpha}$ in relations (5.5) and (5.6), and
$\e^{-1}n^{1/\alpha}\bar L(n)^{1/\alpha}$ instead of $x j^{1/\alpha}\bar
L(j)^{1/\alpha}$ in relations (5.8) and (5.9). The main difference in
the proof of the almost sure functional limit theorem and in the proof
of the limit theorem for the distribution of the normalized partial sum
is that now we make the same truncation for all indices $1\le j\le n$.
Then we can bound the expression we get if we take the sum of the
first two terms in~(5.17) for indices $j=1,2,\dots,n$ by means of
Chebishev's inequality. We get that
$$
\align
P&\(\frac1{n^{1/\alpha}\bar L(n)^{1/\alpha}}\summ_{j=1}^n\( (\xi_j(\oo)-
\eta_j(\oo)) I(|\eta_j(\oo)|<\e^{-1}n^{1/\alpha}\bar
L(n)^{1/\alpha})-a_n\)>\e\)\\
&\hskip7truecm\le\const \e^2.
 \tag5.18
\endalign
$$
On the other hand,
$$
\summ_{j=1}^nP\(|\eta_j(\oo)|\ge\e^{-1}n^{1/\alpha}\bar
L(n)^{1/\alpha}\)\le n\times \const\e^{\alpha}n^{-1}=
\const\e^{\alpha} \tag5.19
$$
by relations (1.5) and (5.3). Relations (5.18) and (5.19) imply the
weak convergence. Theorem~3 is proved. (The functional limit theorem
formulated in Remark~1 can be proved by some modification of the proof,
in particular by applying the Kolmogorov inequality instead of the
Chebishev inequality in the estimates.)
\medskip\noindent{\it Proof of Theorem $3'$.}\/ The stable
process $X_0(t,\oo)$ is not self-similar, because only the relation
$$
X_0(tT,\oo)\overset\Delta\to=TX_0(t,\oo)+\gamma tT\log T
$$
holds with $\gamma=C_1-C_2$, where $\overset\Delta\to=$ denotes
equation in distribution. In this case the process
$X_0'(t,\oo)=X_0(t,\oo)-\gamma t\log t$ is self-similar with
self-similarity parameter $\alpha=1$. Indeed, the relation
$$
\align
X_0'(tT,\oo)&=X_0(tT,\oo)-\gamma tT \log tT\overset\Delta\to=
TX_0(t,\oo)+\gamma tT(\log T-\log tT)\\
&=T(X_0(t,\oo)-\gamma t\log t)=TX'_0(t,\oo)
\endalign
$$
holds for all $T>0$. The results of
Part~I. can be applied for this process. They imply that the random
variables $\eta_n(\oo)=\bar L(n)\bar\eta_n(\oo)-C_n$ satisfy the almost
sure functional limit theorem with $C_n=\gamma(B_n\log B_n-B_{n-1}\log
B_{n-1})$, where $\eta_j(\oo)$, $n=1,2,\dots$, are i.i.d. random
variables with distribution function $G(x)$. Then checking the proof
of Theorem~3 one can see that the estimates given there also hold in
the case $\alpha=1$. They imply that relation (5.4) also holds in this
case. Hence Property~A holds for the pairs $(\xi_n(\oo)-C_n,
\eta_n(\oo)-C_n)$. This result implies that the sequence
$\xi_n(\oo)-C_n$ or the sequence $\xi_n(\oo)$ with an
appropriate (modified) shift $a_n$ and weight function
$B_n=\summ_{k=1}^n\bar L(k)$ satisfies  the almost sure functional
limit theorem.
 
\beginsection 6. Discussion of the results, and some open problems
 
Our results state that for a sequence of random variables
$\xi_n(\oo)$, $n=1,2,\dots$, with some nice properties a result of
the following type holds. Define a sequence of  random
broken lines $X_n(\cdot,\oo)$, $n=1,2,\dots$, in the way
described in formula (1.2) by means of these random variables
$\xi_1(\oo),\xi_2(\oo),\dots$, for all $n=1,2.\dots$, and
$\oo\in\Omega$, define the probability $\mu_N(\oo)$ in the function
space $D([0,1])$ (or in the space $C([0,1])$ if this is possible) by
attaching an appropriately defined probability $a_{k,N}$ to the
appropriately normalized version of the trajectories $X_k(\cdot,\oo)$,
$k=1,\dots,N$, $\summ_{k=1}^N a_{k,N}=1$. Carrying out the above
construction in an appropriate way, e.g.\ in the way described in this
work, we get that the measures $\mu_N(\oo)$ weakly converge to an
appropriate measure $\mu_0$ for almost all $\oo\in\Omega$. In the first
part of this work we proved such results for general self-similar
processes. A weaker version of such results also appeared in the paper
[9] of Cs\'aki and F\"oldes. In the second part we proved such results
for processes which are close to some special self-similar processes.
Actually the transformation which enables one to construct
stationary processes by means of self-similar processes and vice versa
was found already in Lamperti's paper~[17]. This transformation which
enabled us to study self-similar processes by means of ``generalized
Ornstein--Uhlenbeck processes'' was applied by Lamperti to construct
self-similar processes. Let us remark that this method in itself does
not settle the problem of construction of self-similar processes. An
important question is to construct such self-similar processes which
are also stationary or have stationary increments. Such constructions
demand new ideas.
 
One may ask how close these measures $\mu_N(\oo)$, $N=1,2,\dots$, are
to the limit measure $\mu_0$. The following two questions seem to be
natural problems in this direction.
\medskip
\item{i.)} Can a more precise estimate be given about the distance
$d(\mu_N(\oo),\mu_0)$, where $d(\cdot,\cdot)$ is an appropriate
metric on the space of probability measures which metrizes weak
convergence? With which replacement of the weights $\dfrac{\log\frac
{B_{k+1}}{B_k}}{\log\frac{B_N}{B_1}}$ in formula (1.3) do the measures
$\mu_N(\oo)$, $N=1,2,\dots$, have the same limit for almost all $\oo\in
\Omega$ as the original measures in the definition of the almost sure
invariance principle?
\item{ii.)} The weak convergence of the measures $\mu_N(\oo)$
to $\mu_0$ states that $\int \Cal F(x)\mu_N(\oo)(\,dx)\to\int\Cal F(x)
\mu_0(\,dx)$ as $N\to\infty$ for all continuous and bounded functional
$\Cal F$ in the space $D([0,1])$ (or $C([0,1])$. For which larger
classes of functionals $\Cal F$ does this statement hold?
\medskip
There are some results in the spirit of problem~1, see e.g.~[14], but
some further, deeper results in this direction would be welcome. The
second problem is a natural version of the generalization of the Donsker
theorem, and description of the so-called Donsker classes. This is a
popular subject, (see e.g.~[12]), but I do not know of any improvement
of the almost sure functional limit theorem in this direction. Let us
also remark that the basis of our proofs was the application of the
ergodic theorem for the ``generalized Ornstein--Uhlenbeck processes".
In the case of general self-similar processes we cannot assume a
stronger result, but in some important special cases, like in the case
of (non-generalized) Ornstein--Uhlenbeck process there is a chance to
improve the ergodic theorem and to get a non-trivial partial answer to
the question~(i).
 
Another natural generalization of the almost sure invariance principle
is to prove the existence of limit of the appropriate  weighted average
of the variables $\Cal F S_k(\cdot,\oo)$ for almost all $\oo\in \Omega$,
where the trajectories $S_k(\cdot,\oo)$ are defined in formula (1.2),
not only for bounded continuous functionals in the space $C([0,1])$ or
$D([0,1])$, but also for such functionals $\Cal F$ which satisfy
certain moment conditions, but may be unbounded. The idea behind such a
generalization is that the ergodic theorem which is in the background
of the proofs requires only that certain moment condition be satisfied.
 
There are several papers about such problems. These papers consider such
special functionals which depend only on $S_k(1,\oo)$. The deepest
result in this direction I know about is contained in the paper~[16]
of Ibragimov and Lifshitz.
 
Roughly speaking, the second part of this work stated that the almost
sure functional limit theorem holds for independent random variables
under the conditions of the limit theorem for the distribution of the
normalized partial sums of these random variables. Let us also remark
that there are examples (see e.g.~[4]) showing that the almost sure
functional limit theorem also may hold in cases when the limit theorem
for the normalized partial sums of these random variables does not hold.
 
The construction and proofs (the formulation and application of the
Basic Lemma) in Part~II. strongly exploited the independence of the
random variables under consideration. The question arises what can be
said in the dependent case. Does the almost sure limit theorem hold
under general conditions?  Are the conditions sufficient for a limit
theorem for the distribution of the normalized partial sums sufficient
also for the almost sure functional limit theorem? I would formulate
as a `fist rule' a positive answer to this question, but cannot
supply a proof. It might be interesting to study such limit theorems
where the limit is a self-similar process with dependent increments.
The process constructed by Dobrushin in~[10], and the papers proving
limit theorem with this limit (see e.g.\ [11] or~[19]) may be
interesting in this respect.
 
We make a brief comparison between our results and results of earlier
papers. A  more detailed overview together with a comprehensive list
of literature can be found in paper~[3].
 
The most frequently studied problem in this subject is the case when
the limit is Gaussian. Most works is restricted to a
one-dimensional version of the almost sure functional limit theorem
called the almost  sure central limit theorem. This result states
that under general condition the partial sums $S_n(\oo)
=\summ_{j=1}^n\xi_j(\oo)$, $n=1,2,\dots$, of independent random
variables with expectation zero satisfy the relation
$$
\lim_{n\to\infty} \frac1{\summ_{k=1}^n b_k}\sum_{k=1}^n\frac
1{b_k}P\( S_k<x\, \sqrt{\text{Var}\, S_k} \)=\Phi(x)\quad \text{for
almost all }\oo\in\Omega
$$
with the choice of appropriate weights $b_n$, $n=1,2,\dots$, where
$\Phi(\cdot)$ denotes the normal distribution function. The result of
M.~Atlagh~[1] is the sharpest result in  this direction among the
results I know about. These results are very close to ours.
 
We also discuss briefly some analogs of Theorems 2 and 3. There
are results (see e.g. [2],~[5] and [6]) which supply a one-dimensional
version of these results. Here the weight functions are different from
ours. At this point it is important that these results only
deal with  one dimensional distribution and not with the random broken
lines which contain `the whole history' of the process. In these one
dimensional problems some  general theorems about averaging, see e.g.\
the paper Bingham and Roger~[7], give a fairly big freedom in the choice
of the weight functions. On the other hand, in the case of the almost
sure functional limit theorem  a radical change of the weight functions
also  modifies the  points where the random broken lines have a jump,
hence it may modify  the shape of the broken lines. This means that in
such a case we have less freedom in the choice of the weights. Let us
remark that the class of weight  functions for which the almost sure
functional limit theorem holds could be  enlarged. There is a
possibility to generalize the class of possible  weights given in
Theorem~4 of Part~I. to triangular arrays $B_{k,n}$, $1\le k\le n$
under appropriate conditions. This would give a better possibility to
compare our results with those of [2],~[5] and~[6], but we shall not
discuss this problem.
 
Finally we remark that in Theorem~3 (and Theorem~$3'$) we have to give
a `shift parameter'~$a_n$ beside the weight functions $B_n$ to define
the random broken lines which satisfy the almost sure functional limit
theorem. Here again we have certain freedom. As we showed in the
Remark~2. in Section~1 this norming constants can be chosen in the same
way as in the limit theorems for the distribution function of the
normalized partial sums. This means that, by the limit theorems
 with a stable limit law we can choose $a_n=0$ in the case $\alpha<1$
and $a_n=E\xi_1(\oo)$ in the case $1<\alpha<2$. The choice of the
norming constant $a_n$ cannot be given in such a simple way in the
case $\alpha=1$.
 
The results of [5] and of the recent paper [15] also imply Theorem~3
and~$3'$. The method of these papers is different from ours.
 
After having finished this work I have learned about the recent paper
[15] of Ibragimov. ~I.~A.\ and Lifshitz, M.~A.\ which has not yet
appeared. The subject of this paper is similar to ours, but there are
considerable differences both in the formulation of the results and
method of the proofs. The aim of the authors of this work --- similarly
to the present paper --- is to find the general principles and results
in the theory of almost sure limit theorems. They prove both one
dimensional and functional almost sure limit theorems. They show that
the usual limit theorems for distribution functions, which they apply
in an equivalent form expressed by means of characteristic functions,
also imply the almost sure limit theorems. It is worth mentioning that
the proof of the almost sure functional limit theorem in~[15]
formulated in Theorem~3.1 contains an interesting idea which also could
help to considerably simplify the proof of Theorem~1 in Part~I.\ of the
present paper.
 
 
\beginsection REFERENCES
 
\item {1.} Atlagh, M.: Th\'eor\`eme central limite presque s\^ur et
loi du logarithme it\'er\'e pour des sommes de variables al\'eatoires
ind\'ependantes. C.~R.~Acad.~Sci.~Paris S\'er.~I. 316 (1993), 929--933.
 
\item {2.} Berkes, I.:  On the almost sure central limit
theorem and domains of attraction. Probab. Theory Rel.~Fields 102
(1995), 1--18.
 
\item{3.} Berkes. I.: Results and problems related to the pointwise
central limit theorem. In Asymptotic Methods in Probability and
Statistics, Elsevier, (1998) 59--96.
 
\item {4.} Berkes, I.~and Cs\'aki, E.: On the pointwise central limit
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\bye
 
 
