ALMOST SURE FUNCTIONAL LIMIT THEOREMS Part II. The case of independent random variables Péter Major Mathematical Institute of the Hungarian Academy of Sciences and Bolyai College of Eötvös Loránd University, Budapest

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.

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 ξ_{n}(ω), n = 1, 2, …, be a sequence of random variables on a probability space

(Ω, A, P), and let us define the partial sums S_{n}(ω) = ∑_{k=1}^{n} ξ_{k}(ω), n = 1, 2, …, S_{0}(ω) ≡ 0.

Let a monotone increasing sequence B_{n}, n = 0, 1, …, of real numbers be given such that

B_{0} = 0, lim_{n→∞} B_{n} = ∞, and lim_{n→∞} \frac{B_{n+1}}{B_{n}} = 1  (1.1)

together with a positive number α > 0, and define, with the help of the above partial sums S_{n}(ω), n = 1, 2, …, the broken lines

\begin{aligned}S(s, ω) & = S_{j}(ω) if B_{j−1} ≤ s < B_{j}, \\ S_{k}(s, ω) & = B_{k}^{−1/α}S(B_{k}s, ω), 0 ≤ s ≤ 1, \\ & = \frac{S_{j−1}(ω)}{B_{k}^{1/α}} if s_{j−1,k} ≤ s < s_{j,k}, 1 ≤ j ≤ k, S_{k}(1, ω) = \frac{S_{k}(ω)}{B_{k}^{1/α}}, \\ & k = 1, 2, …, \end{aligned} (1.2)

where s_{j,k} = \frac{B_{j}}{B_{k}}, 0 ≤ j ≤ k. Now we introduce the following definition.

Definition of the almost sure functional limit theorem. Let ξ_{n}(ω), n = 1, 2, …, be a sequence of random variables, and let a sequence of real numbers B_{n}, n = 1, 2, …, be given which satisfies formula (1.1) together with some α > 0. Let us consider the random broken lines S_{k}(s, ω), 0 ≤ s ≤ 1, defined with the help of their partial sums

S_{k}(ω) = ∑_{j=1}^{k} ξ_{j}(ω), k = 1, 2, …, by formula (1.2). For all ω ∈ Ω and N = 1, 2, … define the random probability measures μ_{N}(ω) on the space D([0, 1]) of càdlàg (continuous from the right, limit from the left) functions on the interval [0, 1] in the following way: The measure μ_{N}(ω) is concentrated on the random broken lines S_{k}(·, ω), 1 ≤ k ≤ N, defined in formula (1.2), and

μ_{N}(ω)(S_{k}(·, ω)) = \frac{1}{log \frac{B_{N}}{B_{1}}} log \frac{B_{k+1}}{B_{k}} , 1 ≤ k < N.  (1.3)

The sequence of random variables ξ_{n}(ω), n = 1, 2, …, satisfies the almost sure functional limit theorem with weight function B_{n}, n = 1, 2, …, parameter α > 0 and limit measure μ_{0} on the space D([0, 1]) if for almost all ω ∈ Ω the probability measures μ_{N}(ω) defined with the help of the above constants B_{n} and α (appearing in formula (1.2)) converge weakly to the measure μ_{0} as N → ∞.

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 μ_{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 α. But since in all cases we prove the almost sure functional limit theorem a sequence of the form A_{n} = B_{n}^{1/α} 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, ω), t ≥ 0, be given with a self-similarity parameter α > 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, …, of real numbers for which relation (1.1) holds. Then the random variables η_{n}(ω) = X(B_{n}, ω) − X(B_{n−1}, ω), n = 1, 2, …, satisfy the almost sure functional limit theorem with weight function B_{n}, n = 1, 2, …, parameter α and limit measure μ_{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, ω) 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 η_{n}(ω), n = 1, 2, …, Eη_{n}(ω) = 0, Eη_{n}^{2}(ω) = B_{n} − B_{n−1} with the weight functions B_{n}, parameter α = 2 and limit measure μ_{0} which is the Wiener measure, i.e. the distribution of the process W (t, ω), 0 ≤ t ≤ 1. Similarly, we get by considering a stable process X(t, ω) with self-similarity parameter α, 0 < α < 2, α ≠ 1, as a self-similar process and a sequence of real numbers B_{n} satisfying relation (1.1), that a sequence of independent random variables η_{n}(ω), n = 1, 2, …, such that the distribution of η_{n}(ω) agrees with the distribution of X(B_{n} − B_{n−1}, ω) satisfies the almost sure functional limit theorem with weight function B_{n}, parameter α and limit measure μ_{0} which is the distribution of the restriction of the process X(t, ω) 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.

Theorem 1. Let ξ_{n}(ω), n = 1, 2, …, be a sequence of independent random variables such that Eξ_{n}(ω) = 0, Eξ_{n}^{2}(ω) = σ_{n}^{2}, and it satisfies the Lindeberg condition, i.e.

lim_{n→∞} \frac{1}{D_{n}^{2}} ∑_{k=1}^{n} Eξ_{k}^{2}(ω)I(|ξ_{k}(ω)| ≥ εD_{n}) = 0 for all ε > 0,  (1.4)

where D_{n}^{2} = ∑_{k=1}^{n} σ_{k}^{2}. Then the sequence of random variables ξ_{n}(ω), n = 1, 2, …, satisfies

the almost sure functional (central) limit theorem with the Wiener measure μ_{0} as the limit measure, weight function B_{n} = D_{n}^{2}, n = 1, 2, …, and parameter α = 2.

Theorem 2. Let ξ_{n}(ω), n = 1, 2, …, be a sequence of independent, identically distributed random variables with a non-degenerated distribution such that Eξ_{1}(ω) = 0 and μ(x) = Eξ_{1}^{2}(ω)I(|ξ_{1}(ω)| ≤ x) is a slowly varying function at infinity. Define the num-

bers a_{n} as a_{n} = sup \{u: n\frac{μ(u)}{u^{2}} ≥ 1\} for all n ≥ n_{0} with a sufficiently large integer n_{0}.

(This definition is meaningful. To see this observe that n\frac{μ(u)}{u^{2}} > 1 for an appropriate

u and all sufficiently large n, and lim_{u→∞} \frac{nμ(u)}{u^{2}} = 0 for all n if μ(u) is a slowly varying function.) Define the sequence a_{n} in the above way if n ≥ n_{0}, and for the sake of a unique definition put a_{n} = a_{n_{0}} for n ≤ n_{0}. Then the sequence ξ_{n}(ω), n = 1, 2, …, satisfies the almost sure (central) functional limit theorem with the Wiener measure μ_{0} as

the limit measure, weight function B_{n} = ∑_{k=1}^{n} μ(a_{k}), n = 1, 2, …, and parameter α = 2.

To formulate the following Theorem 3 let us recall that the distribution function G(x) of a stable law, 0 < α < 2, is determined by three parameters α, C_{1} ≥ 0 and C_{2} ≥ 0, C_{1} + C_{2} > 0, such that the relations

\begin{aligned}G(x) & ∼ C_{1}x^{−α} if x → ∞, \\ 1 − G(−x) & ∼ C_{2}x^{−α} if x → ∞ \end{aligned}  (1.5)

hold. Let us also recall that for all stable laws G(x) satisfying (1.5) with 0 < α < 2, α ≠ 1, there is a (stable) process X(t, ω), 0 ≤ t ≤ 1, with trajectories in the space D([0, 1]) such that it has independent and stationary increments, and X(1, ω) has the distribution function G(x). The distribution of this process X(t, ω) in the space D([0, 1]) is uniquely determined. It is a self-similar process with self-similarity parameter α.

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:

\begin{aligned}1 − F (x) & ∼ C_{1}x^{−α}L(x) \\ F (−x) & ∼ C_{2}x^{−α}L(x) \end{aligned}  if x → ∞,  (1.6)

where L(x) is a slowly varying function at infinity, C_{1} ≥ 0, C_{2} ≥ 0, C_{1} + C_{2} > 0, 0 < α < 2, α ≠ 1. Let us define the following functions b(x) and L̄(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 \{u: \frac{L(u)x}{u^{α}} ≥ 1\}, and let L̄(x) = b(x)^{α}x^{−1}, where the

number α and function L(·) are the same as in formula (1.6). Let μ_{0} denote the (uniquely determined) distribution of the stable process X(t, ω) with parameter α > 0, 0 ≤ t ≤ 1, in the space D([0, 1]) for which X(1, ω) has that stable distribution G(x) which satisfies relation (1.5). (The numbers α, C_{1} and C_{2} are the same in formulas (1.5) and (1.6).)

Theorem 3. Let ξ_{n}(ω), n = 1, 2, …, 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 ξ_{n}(ω)−a_{n}, n = 1, 2, …, satisfies the almost sure functional limit theorem with

weight function B_{n} = ∑_{k=1}^{n} L̄(k), n = 1, 2, … and limit measure μ_{0}, where the function

L̄(x) and measure μ_{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/α, hence L̄(x) = b(x)^{α}x^{−1} is a slowly varying function.

The normalized partial sums \frac{1}{B_{n}^{1/α}} ∑_{k=1}^{n}(ξ_{k}(ω)−a_{n}) converge in distribution to G(x)

as n → ∞ if the constants a_{n} are the same as in the almost sure functional limit theorem formulated in this theorem.

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, … from the normalized

partial sum S_{k}(ω) = \frac{1}{B_{n}^{1/α}} ∑_{j=1}^{k}(ξ_{j}(ω) − a_{n}), k = 1, …, n, in the natural way. The limit

is the same measure μ_{0} which appears as the limit in the almost sure limit theorem.

We also explain without working out all details that Theorem 3 also holds for α = 1 with some slight natural modifications.

Theorem 3′. The results of Theorem 3 also hold in the case α = 1 with the slight modification that in this case the “shift parameters” a_{n}, n = 1, 2, … must be defined differently, the measure μ_{0} is the distribution of the process X(t) − γt log t with the constant γ = γ(C_{1}, C_{2}) = C_{1} − C_{2}, where X(t, ω) is the stable process with parameter α = 1 for which X(1, ω) has distribution function G(x) which satisfies relation (1.5).

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 ā_{n} if and only if a_{n} − ā_{n} = o (\frac{B_{n}^{1/α}}{n} ). This implies that the norming

sequence a_{k}, k = 1, 2, …, in the almost sure functional limit theorem can be replaced by any such sequence ā_{k} for which the number ā_{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 lim_{n→∞} ∑_{k=1}^{n} \frac{a_{k} − ā_{k}}{B_{n}^{1/α}} = 0, since

∑_{k=1}^{n} \frac{a_{k} − ā_{k}}{B_{n}^{1/α}} = o ( ∑_{k=1}^{n} \frac{B_{k}^{1/α}}{kB_{n}^{1/α}} ) = o(1).

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 (ξ_{n}(ω), η_{n}(ω)), n = 1, 2, …, and the sequence η_{n}(ω), n = 1, 2, …, satisfies the almost sure functional limit theorem, then the sequence of random variables ξ_{n}(ω), n = 1, 2, …, also satisfies the almost sure functional limit theorem with the same weight function B_{n}, parameter α and limit measure μ_{0} as the sequence η_{n}(ω), n = 1, 2, . . .. 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.

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.

Definition of Property A. Let η_{n}(ω), n = 1, 2, …, be a sequence of random variables which satisfies the almost sure functional limit theorem with a limit measure μ_{0} in the space D([0, 1]) with some weight function B_{n}, n = 0, 1, …, satisfying relation (1.1) and parameter α > 0. Let us also assume that the limit measure μ_{0} is the distribution of the restriction of a self-similar process X(u, ω) with self-similarity parameter α > 0 to the interval 0 ≤ u ≤ 1.

Define the indices N(n) as N(n) = inf{k: B_{k} ≥ 2^{n}}, n = 0, 1, . . .. The pair of sequences of random variables (ξ_{n}(ω), η_{n}(ω)), n = 1, 2, …, satisfies Property A if for all ε > 0 there exists a sequence of random variables ξ̃_{n}(ω) = ξ̃_{n}(ε, ω), n = 1, 2, …, whose (joint) distribution agrees with the (joint) distribution of the sequence ξ_{n}(ω), n =

1, 2, …, and the partial sums S̃_{n}(ω) = ∑_{k=1}^{n} ξ̃_{k}(ω) and T_{n}(ω) = ∑_{k=1}^{n} η_{k}(ω), n = 1, 2, …,

satisfy the relation

lim sup_{n→∞} \frac{1}{n} ∑_{k=1}^{N(n)} log \frac{B_{k+1}}{B_{k}} I (\{ω: \frac{sup_{0≤j≤k} |S̃_{j}(ω) − T_{j}(ω)|}{B_{k}^{1/α}} > ε\}) ≤ ε  (2.1)

for almost all ω ∈ Ω, where I(A) denotes the indicator function of the set A.

In Theorem 4 of Part I. we have proved that if the pair of sequences of random variables (ξ_{n}(ω), η_{n}(ω)), n = 1, 2, …, satisfies Property A, then the sequence of random variables ξ_{n}(ω), n = 1, 2, …, satisfies the almost sure functional limit theorem with the same weight function B_{n}, parameter α > 0 and limit measure μ_{0} as the sequence of random variables η_{n}(ω), n = 1, 2, . . .. 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 (ξ̃_{n}(ω), η_{n}(ω)), n = 1, 2, . . .. 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 S̃_{n}(ω) − T_{n}(ω), n = 1, 2, …, made from these random variables are relatively small for almost all ω.

Let us choose an appropriate subsequence n_{j} of the integers and apply a socalled quantile transform, to be described later, which makes the differences (S_{n_{j}}(ω) − S_{n_{j−1}}(ω))−(T_{n_{j}}(ω)−T_{n_{j−1}}(ω)) relatively small, and for which the expressions S_{n_{j}}(ω)− S_{n_{j−1}}(ω) and T_{n_{j}}(ω)−T_{n_{j−1}}(ω) have the right distribution. We make such constructions independently for all j = 1, 2, …, and define the random variables S_{n_{j}}(ω) and T_{n_{j}}(ω) as the partial sums of these terms. Then these subsequences can be extended to two sequences S_{n}(ω) and T_{n}(ω) which have the same joint distribution as the partial sums of the independent random variables ξ_{k}(ω) and η_{k}(ω), k = 1, 2, . . .. The differences between the random variables S_{n_{j}}(ω) − S_{n_{j−1}}(ω) and T_{n_{j}}(ω) − T_{n_{j−1}}(ω) 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}}(ω)−T_{n_{j}}(ω), and then by bounding the fluctuation of the sequences S_{n}(ω) and T_{n}(ω) 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}(ω) of partial sums of independent random variables with a sequence S_{n}(ω), n = 1, 2, …, 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}(ω) − T_{n}(ω) be small for most indices n. The existence of some exceptional indices n (depending on ω) 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 ζ_{k}(ω), k = 1, 2, …, if they satisfy certain conditions. The bound (2.6) proved in the Basic Lemma will be applied with the choice ζ_{k}(ω) = ξ̃_{k}(ω) − η_{k}(ω) where ξ̃_{k}(ω) and η_{k}(ω), k = 1, 2, …, are random variables constructed in the above way. Let us remark that when the sequences S_{n}(ω) and T_{n}(ω) 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, ω) 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)} ∼ 2^{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 ζ_{k}(ω) in formula (2.3) of the Basic Lemma which will be satisfied in our applications with the natural choice ζ_{k}^{(1)}(ω) = ξ̃_{k}(ω) and ζ_{k}^{(2)}(ω) = η_{k}(ω). The Basic Lemma also contains a condition about the independence of the random variables U_{N(n,k)}(ω) − U_{N(n,k−1)}(ω), n = 1, 2, …, k = 1, …, 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 ξ̃_{j}(ω) and η_{j}(ω) 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 ξ_{k}(ω) and η_{k}(ω), k = 1, 2, . . .. 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 ε > 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)} ∼ ε̄2^{n}, where the coefficient ε̄ > 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 ξ_{n}(ω) and η_{n}(ω) 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 ξ_{n}(ω) and η_{n}(ω), 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 ξ_{n}(ω) 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.

Definition of refinement of a sequence of integers. Given a sequence 0 = N(0) < N(1) < N(2) < ⋯ of integers we call the refinement of this sequence N(n), n = 0, 1, 2, … a set of non-negative integers N(n, k) indexed by two parameters n = 1, 2, …, and 0 ≤ k ≤ l_{n} with some positive integer l_{n} such that

N(n − 1) = N(n, 0) < N(n, 1) < ⋯ < N(n, l_{n}) = N(n), n = 1, 2, ….

Basic Lemma. Let B_{n}, n = 0, 1, …, B_{0} = 1, be a sequence of real numbers which satisfies relation (1.1). Let a sequence of random variables ζ_{n}(ω), n = 1, 2, … and a number α > 0 also be given. Define the sequence N(n) = inf{k: B_{k} ≥ 2^{n}}, n = 1, 2, …, N(0) = 0. We give an estimate on the maximum of the partial sums of the random variables ζ_{n}(ω) under appropriate conditions.

Fix a number ε > 0 and a refinement N(n, k), n = 0, 1, …, 0 ≤ k ≤ l_{n}, of the sequence N(n), n = 1, 2, …, which may depend on ε. Put

U_{n}(ω) = ∑_{j=1}^{n} ζ_{j}(ω), n = 1, 2, …,

and

V_{n}(ω) = sup_{0<j≤l_{n}} |U_{N(n,j)}(ω) − U_{N(n,0)}(ω)| , n = 1, 2, ….  (2.2)

Let us assume that there is a decomposition

ζ_{k}(ω) = ζ_{k}^{(1)}(ω) − ζ_{k}^{(2)}(ω), k = 1, 2, …  (2.3)

of the random variables ζ_{k}(ω) in such a way that both sequences ζ_{k}^{(i)}(ω), k = 1, 2, …, i = 1, 2, consist of independent random variables which satisfy certain inequalities formulated below. To formulate them let us introduce the notation

ζ_{n,k,m}^{(i)}(ω) = ζ_{N(n,k−1)+m}^{(i)}(ω), n = 1, 2, …, k = 1, …, l_{n},
1 ≤ m ≤ N(n, k) − N(n, k − 1), i = 1, 2

with the help of the refinement N(n, k) of the sequence N(n), n = 1, 2, …, fixed in this lemma. Let us assume that the following inequalities hold:

P (  sup_{1≤p<N(n,k)−N(n,k−1)} | ∑_{m=1}^{p} ζ_{n,k,m}^{(i)}(ω)| > εx2^{n/α}) ≤ \frac{C_{1}εx^{−γ}}{l_{n}}
i = 1, 2, for all x ≥ 1, n ≥ n_{0}, and 1 ≤ k ≤ l_{n}

(2.4)

with some constants n_{0} = n_{0}(ε) > 0, γ > 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)}(ω) − U_{N(n,k−1)}(ω), n = 1, 2, …, k = 1, …, l_{n}, are independent, and the inequality

P (V_{n}(ω) ≥ εx2^{n/α}) ≤ C_{2}εx^{−γ} for all x ≥ 1 and n ≥ n_{0}  (2.5)

holds with some n_{0} = n_{0}(ε), γ > 0 and C_{2} > 0. Then

lim sup_{n→∞} \frac{1}{n} ∑_{k=1}^{N(n)} log \frac{B_{k+1}}{B_{k}} I (\frac{sup_{0≤s≤k} |U_{s}(ω)|}{B_{k}^{1/α}} > Kε) ≤ Kε

for almost all ω ∈ Ω.

(2.6) with an appropriate constant K = K(C_{1}, C_{2}, γ, α) > 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 ε.)

We shall be able to prove Property A under weak conditions by applying the Basic Lemma for arbitrary small ε > 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}(ω) − T_{n}(ω) be small in some average. Let us remark that there are even results, (see Berkes and Csáki [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|^{−γ} at the right-hand side of these formulas could be replaced by (1 + log |x|)^{−γ′} with a sufficiently large γ′ > 0. But such a condition does not seem to be better applicable in the problems we are interested in.

3. Proof of the Basic Lemma

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} = ∑_{k=1}^{p} l_{k}, p = 1, 2, …, and the se-

quence m(j), j = 0, 1, …, by the formula m(0) = 0 and m(j) = N(p − 1, j − L_{p−1}), if L_{p−1} < j ≤ 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, . . .. Put A_{n} = B_{n}^{1/α}. We shall prove that

lim sup_{n→∞} \frac{1}{n} ∑_{j=1}^{L_{n}} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}} I (\frac{sup_{1≤s≤j} |U_{m(s)}(ω)|}{A_{m(j)}} > K_{1}ε) ≤ K_{1}ε

for almost all ω ∈ Ω

(3.1)

with an appropriate constant K_{1} > 0, and

lim sup_{n→∞} \frac{1}{n} ∑_{j=1}^{L_{n}} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}} I (\frac{sup_{1≤s≤j} sup_{m(s−1)<p<m(s)} |U_{p}(ω) − U_{m(s−1)}(ω)|}{A_{m(j)}} > K_{2}ε)

≤ K_{2}ε for almost all ω ∈ Ω

(3.2)

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/α}(K_{1} + K_{2}).

Indeed, if for some ω ∈ Ω there is an index k, n_{0} ≤ k ≤ N(n) with some n_{0} = n_{0}(ε), such that it gives a non-zero contribution to the sum in (2.6) with the choice K = 3^{1/α}(K_{1} +K_{2}), i.e. sup_{1≤s≤k} |U_{s}(ω)| > 3^{1/α}(K_{1} +K_{2})A_{k}ε, then consider that interval

(m(j − 1), m(j)], 1 ≤ j ≤ L_{n}, which contains this number k. In this case one of the following relations holds. Either

sup_{1≤s≤j} |U_{m(s)}(ω)| > 3^{1/α}K_{1}A_{k}ε ≥ K_{1}A_{m(j)}ε

or

sup_{1≤s≤j} sup_{m(s−1)<p<m(s)} |U_{p}(ω) − U_{m(s−1)}(ω)| > 3^{1/α}K_{2}A_{k}ε ≥ K_{2}A_{m(j)}ε.

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 \frac{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 ≤ j ≤ L_{n} imply formula (2.6).

To prove relation (3.1) introduce the random variables

T_{s}(ω) = U_{m(s)}(ω) − U_{m(s−1)}(ω), s = 1, 2, ….

The random variables T_{s}(ω), s = 1, 2, … are independent. This statement is equivalent to the independence of the random variables U_{N(n,k)}(ω) − U_{N(n,k−1)}(ω), n = 1, 2, …, k = 1, …, l_{n}, and this is a condition imposed in the formulation of the Basic Lemma.

Since lim_{n→∞} \frac{B_{N(n+1)}}{B_{N(n)}} = 2, A_{n} = B_{n}^{1/α}, there is some n_{0} > 0 such that

A_{N(n)} ≥ 2^{(n−k)/2α}A_{N(k)} for arbitrary n ≥ n_{0} and k ≤ n.

For all s = 1, 2, … define the number R(s) which satisfies the inequality L_{R(s)−1} < s ≤ 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) ≤ 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)}} ≥ 2^{(R(j)−R(s)−1)/2α} for 1 ≤ s ≤ j and j ≥ n_{0}.

Let us fix some j ≥ n_{0}. We shall show by applying the above relation for s ≤ j and by putting in one block those indices s for which N(r − 1) < m(s) ≤ N(r), or equivalently L_{r−1} < s ≤ L_{r} that

\{ω: \frac{sup_{1≤s≤j} |U_{m(s)}(ω)|}{A_{m(j)}} ≥ K_{1}ε\} = \{ω: |∑_{p=1}^{s} T_{p}(ω)| ≥ K_{1}εA_{m(j)} for some 1 ≤ s ≤ j\}

⊂ ⋃_{r=1}^{R(j)} {ω: sup_{L_{r−1}<u≤L_{r}} | ∑_{p=L_{r−1}+1}^{u} T_{p}(ω)| ≥ CK_{1}ε2^{(R(j)−r)/4α}A_{m(L_{r−1})}}

(3.3)

with C = 1 − 2^{−1/4α}. To prove relation (3.3) observe that if L_{r−1} < j, then

A_{m(j)} ≥ 2^{(R(j)−R(L_{r−1})−1)/2α}A_{m(L_{r−1})} = 2^{(R(j)−r)/2α}A_{m(L_{r−1})},

hence if some ω is not contained in the set at the right-hand side, i.e.

sup_{L_{r−1}<u≤L_{r}} | ∑_{p=L_{r−1}+1}^{u} T_{p}(ω)| < CK_{1}ε2^{(R(j)−r)/4α}A_{m(L_{r−1})} for all 1 ≤ r ≤ R(j)

then

|∑_{p=1}^{s} T_{p}(ω)| < ∑_{r=1}^{R(j)} sup_{L_{r−1}<u≤L_{r}} | ∑_{p=L_{r−1}+1}^{u} T_{p}(ω)|

≤ CK_{1}ε ∑_{r=1}^{R(j)} 2^{−(R(j)−r)/4α}2^{(R(j)−r)/2α}A_{m(L_{r−1})}

≤ CK_{1}ε ∑_{r=1}^{R(j)} 2^{−(R(j)−r)/4α}A_{m(j)} < K_{1}εA_{m(j)}

for all s ≤ L_{R(j)}, hence for all s ≤ j, and this means that ω 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}(ω) introduced in formula (2.2) and the relation A_{m(L_{r−1})} = B_{m(L_{r−1})}^{1/α} = B_{N(r−1)}^{1/α} ≥ 2^{(r−1)/α} that

\{ω: \frac{sup_{1≤s≤j} |U_{m(s)}(ω)|}{A_{m(j)}} ≥ K_{1}ε\} ⊂ ⋃_{r=1}^{R(j)} \{ω: V_{r}(ω) ≥ CK_{1}ε2^{(R(j)−r)/4α} × 2^{(r−1)/α}\}

= ⋃_{r=1}^{R(j)} {ω: V_{r}(ω) ≥ CK_{1}2^{(R(j)+3r−4)/4α}ε}  (3.4)

We shall prove relation (3.1) with the help of (3.4). Let us first sum for R(j) = p with a fixed p ≥ n_{0}. (Observe that the right-hand side of (3.4) depends on j only through R(j).) We get that

∑_{j: R(j)=p} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}} I (\frac{sup_{1≤s≤j} |U_{m(s)}(ω)|}{A_{m(j)}} > K_{1}ε)

≤ ∑_{j: R(j)=p} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}} ∑_{r=1}^{p} I (V_{r}(ω) ≥ CK_{1}2^{(p+3r−4)/4α}ε)

= ∑_{r=1}^{p} I (V_{r}(ω) ≥ CK_{1}2^{(p+3r−4)/4α}ε) ∑_{j: R(j)=p} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}}  (3.5)

≤ log \frac{B_{N(p+1)+1}}{B_{N(p)}} ∑_{r=1}^{p} I (V_{r}(ω) ≥ CK_{1}2^{(p+3r−4)/4α}ε)

≤ 2 ∑_{r=1}^{p} I (V_{r}(ω) ≥ CK_{1}2^{(p+3r−4)/4α}ε)

if p ≥ 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, …, exploiting that the terms of (3.1) not considered in such a way give only a bounded contribution, and the relation R(j) ≤ n holds for j ≤ L_{n}. In such a way we obtain that

∑_{j=1}^{L_{n}} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}} I (\frac{sup_{1≤s≤j} |U_{m(s)}(ω)|}{A_{m(j)}} > K_{1}ε)

≤ 2 ∑_{p=1}^{n} ∑_{r=1}^{p} I (V_{r}(ω) ≥ CK_{1}2^{(p+3r−4)/4α}ε) + const.

(3.6)

Define the random variables

χ_{r}(ω) = ∑_{p=0}^{∞} I (2^{−r/α}V_{r}(ω) ≥ CK_{1}2^{(p−4)/4α}ε) , r = 1, 2, ….

We can write with the help of relation (3.6) by changing the order of summation at the right-hand side that

lim sup_{n→∞} \frac{1}{n} ∑_{j=1}^{L_{n}} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}} I (\frac{sup_{1≤s≤j} |U_{m(s)}(ω)|}{A_{m(j)}} > K_{1}ε)

≤ lim sup_{n→∞} \frac{2}{n} ∑_{r=1}^{n} ∑_{p=r}^{n} I (V_{r}(ω) ≥ CK_{1}2^{r/α}2^{(p−r−4)/4α}ε)  (3.7)

= lim sup_{n→∞} \frac{2}{n} ∑_{r=1}^{n} ∑_{p=0}^{n−r} I (2^{−r/α}V_{r}(ω) ≥ CK_{1}2^{(p−4)/4α}ε) ≤ lim sup_{n→∞} \frac{2}{n} ∑_{r=1}^{n} χ_{r}(ω).

The random variables χ_{r}(ω), r = 1, 2, …, are independent, the relations 0 ≤ Eχ_{r}(ω) ≤ Kε and Eχ_{r}^{2}(ω) ≤ const. hold for r ≥ n_{0}(ε) because of relation (2.5), where an explicit bound can be given for K = K(α, γ). Indeed, the random variables χ_{r}(ω) take non-negative integer values, the set {ω: χ_{r}(ω) ≥ k} agrees with the set {ω: 2^{−r/α}V_{r}(ω) ≥ CK_{1}2^{(k−4)/4α}ε} whose probability can be bounded by C_{1}ε2^{−kγ/4α} by formula (2.5). (We may assume that K_{1} > 0 is chosen so large that CK_{1}2^{−1/α} > 1.) This implies that P (χ_{r}(ω) ≥ x) ≤ C_{1}ε2^{−γx/4α} for r ≥ n_{0}. Hence the laws of large numbers can be applied for these random variables, and we get that

lim_{n→0} \frac{1}{n} ∑_{r=1}^{n}(χ_{r}(ω) − Eχ_{r}(ω)) = 0, for almost all ω ∈ Ω

and lim sup_{n→∞} \frac{1}{n} ∑_{r=1}^{n} Eχ_{r}(ω) ≤ Kε.

These estimates together with relation (3.7) imply (3.1).

To prove relation (3.2) let us introduce the random variables

Z_{n}^{(i)}(ω) = \frac{1}{2^{n/α}} sup_{1≤k≤l_{n}} sup_{1≤p<N(n,k)−N(n,k−1)} | ∑_{m=1}^{p} ζ_{n,k,m}^{(i)}(ω)| ,
n = 1, 2, …, i = 1, 2.

Condition (2.4) implies that

P (Z_{n}^{(i)}(ω) ≥ εx) ≤ C_{1}εx^{−γ} for all x ≥ A n ≥ n_{0}(ε) and i = 1, 2.  (3.8)

We claim that

∑_{j=L_{r−1}+1}^{L_{r}} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}} I (\frac{sup_{1≤s≤j} sup_{m(s−1)<p<m(s)} |U_{p}(ω) − U_{m(s−1)}(ω)|}{A_{m(j)}} > K_{2}ε)

≤ 2 ∑_{u=1}^{r} (I (Z_{u}^{(1)}(ω) > \frac{K_{2}ε2^{(r−u−1)/α}}{2} ) + I (Z_{s}^{(2)}(ω) > \frac{K_{2}ε2^{(r−u−1)/α}}{2} ))

for all r ≥ 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

∑_{j=L_{r−1}+1}^{L_{r}} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}} ≤ 2, and one of the summands at the right-hand side is non-

zero, since Z_{u}^{(1)}(ω) + Z_{u}^{(2)}(ω) > 2^{−u/α}K_{2}εA_{m(j)} ≥ K_{2}ε2^{(r−u−1)/α} for some 1 ≤ u ≤ r. Hence the inequality also holds in this case. By summing up this inequality for r = 1, …, n we get the following bound for the expression in (3.2):

lim sup_{n→∞} \frac{1}{n} ∑_{j=1}^{L_{n}} log \frac{B_{m(j)+1}}{B_{m(j−1)+1}} I (\frac{sup_{1≤s≤j} sup_{m(s−1)<p<m(s)} |U_{p}(ω) − U_{m(s−1)}(ω)|}{A_{m(j)}} > K_{2}ε)

≤ lim sup_{n→∞} \frac{2}{n} ∑_{r=1}^{n} ∑_{u=1}^{r}(I (Z_{u}^{(1)}(ω) > \frac{K_{2}ε2^{(r−u−1)/α}}{2} )  (3.9)

+ I (Z_{u}^{(2)}(ω) > \frac{K_{2}ε2^{(r−u−1)/α}}{2} )).

Let us define the random variables

X_{u}^{(i)}(ω) = ∑_{p=0}^{∞} I (Z_{u}^{(i)}(ω) ≥ \frac{K_{2}ε}{4}2^{(p−1)/α}) , u = 0, 1, 2, …, 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

lim sup_{n→∞} \frac{2}{n} ∑_{u=1}^{n} (X_{u}^{(1)}(ω) + X_{u}^{(2)}(ω)) .

The averages of the random variables X_{u}^{(1)}(ω) + X_{u}^{(2)}(ω) − EX_{u}^{(1)}(ω) − EX_{u}^{(2)}(ω) tend to zero with probability one. Indeed, the random variables X_{u}^{(i)}(ω) 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) ≤ C_{2}2^{−γx/α}, i = 1, 2, u ≥ 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 (χ_{r}(ω) > x) made after formula (3.7).) Moreover, EX_{u}^{(i)}(ω) ≤ Kε for all u ≥ n_{0}(ε) and i = 1, 2, with an appropriate constant K > 0, and as a consequence

lim sup_{n→∞} \frac{2}{n} ∑_{u=1}^{n} (EX_{u}^{(1)}(ω) + EX_{u}^{(2)}(ω)) ≤ 4Kε.

These relations imply formula (3.2). The Basic Lemma is proved.

4. The proof of Theorems 1 and 2

Proof of Theorem 1. Let η_{n}(ω), n = 1, 2, …, be a sequence of independent Gaussian random variables such that Eη_{n}(ω) = 0 and Eη_{n}^{2}(ω) = σ_{n}^{2}. Let us fix a number ε > 0. We want to construct a sequence of independent random variables ξ̃_{n}^{(ε)}(ω), n = 1, 2, …, which has the same distribution as the sequence ξ_{n}(ω), n = 1, 2, …, and the sequences

ζ_{n}(ω) = ξ̃_{n}^{(ε)}(ω) − η_{n}(ω), and U_{n}(ω) = ∑_{j=1}^{n} ζ_{n}(ω), n = 1, 2, …, satisfy relation (2.6)

with B_{n} = D_{n}^{2} = ∑_{k=1}^{n} σ_{k}^{2}, α = 2 and the number ε we have fixed. This relation will

be proved with an application of the Basic Lemma. If we can do this for arbitrary ε > 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 η_{n}(ω), n = 1, 2, …, imply Theorem 1.

We shall omit the sign “˜ ” and “^{(ε)}” and write ξ_{n} instead of ξ̃_{n}^{(ε)}. To apply the Basic Lemma we have to define some quantities. We fix a sufficiently small ε̄ = ε̄(ε) > 0 to be defined later and define the numbers N(n), n = 1, 2, …, by means of the sequence B_{n} = D_{n}^{2} as in the formulation of the Basic Lemma. Then we define an “ε̄ regular refinement” N(n, k), n = 1, 2, …, 0 ≤ k ≤ l_{n}, of the sequence N(n). By this regularity property we mean that

ε̄(B_{N(n)} − B_{N(n−1)}) ≤B_{N(n,k)} − B_{N(n,k−1)} ≤ 3ε̄(B_{N(n)} − B_{N(n−1)})
for n ≥ n_{0}(ε̄) and all 1 ≤ k ≤ l_{n}.

(4.1)

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)} > 3ε̄(B_{N(n)} − B_{N(n−1)}), then

N(n, k + 1) = min{j: B_{j} − B_{N(n,k)} ≥ ε̄(B_{N(n)} − B_{N(n−1)})}.

If B_{N(n)} −B_{N(n,k)} ≤ 3ε̄(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, …, satisfies

relation (1.1), and lim_{N→∞} sup_{N(n−1)≤k≤N(n)} \frac{σ_{k}^{2}}{B_{N(n)}} = 0. Hence lim_{n→∞} 2^{−n}B_{N(n)} = 1, and

B_{N(n,k)} − B_{N(n,k−1)} ∼ ε̄(B_{N(n)} − B_{N(n−1)}), if 1 ≤ k ≤ l_{n} − 1. It is not difficult to see that the sequence N(n, k) is an ε̄ regular refinement of the sequence N(n).

Let F_{n,k}(x) = P (S_{n,k}(ω) < x) denote the distribution function of S_{n,k}(ω) =

\frac{1}{Ā_{n,k}} ∑_{j=N(n,k−1)+1}^{N(n,k)} ξ_{j}(ω) with Ā_{n,k}^{2} = ∑_{j=N(n,k−1)+1}^{N(n,k)} σ_{j}^{2} = B_{N(n,k)} −B_{N(n,k−1)}, and define

the random variables T_{n,k}(ω) = \frac{1}{Ā_{n,k}} ∑_{j=N(n,k−1)+1}^{N(n,k)} η_{j}(ω), n = 1, 2, …, k = 1, …, l_{n}.

Let Φ(x) denote the standard normal distribution function. Then T_{n,k}(ω), n = 1, 2, …, k = 1, …, l_{n}, are independent standard normal random variables, and the variables χ_{n,k}(ω) = Φ(T_{n,k}(ω)) are independent random variables, uniformly distributed in the interval [0, 1].

We shall construct the random variables S_{n,k}(ω) = \frac{1}{Ā_{n,k}} ∑_{j=N(n,k−1)+1}^{N(n,k)} ξ_{j}(ω), n =

1, 2, …, 1 ≤ k ≤ l_{n}, by means of the so-called quantile transform as S_{n,k}(ω) = F_{n,k}^{−1}(χ_{n,k}(ω)), 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}(ω) = G_{n,k}(χ_{n,k}(ω)), n = 1, 2, …, k = 1, …, l_{n}. Such a definition is meaningful for all distribution functions. The random variables S_{n,k}(ω), n = 1, 2, …, k = 1, …, 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}(ω) is really F_{n,k}(x) let us first observe that

P (S_{n,k}(ω) < x) = lim_{h: h>0,h→0} P (S_{n,k}(ω) < x − h)

= lim_{h: h>0,h→0} P (G_{n,k}(χ_{n,k}(ω)) < x − h) ≤ P (χ_{n,k}(ω) ≤ F (x)) = F (x),

since {ω: G(χ_{n,k}(ω)) < x−h} ⊂ {ω: χ_{n,k}(ω) ≤ F (x)} for all h > 0. To see the estimate from the opposite direction observe that P (S_{n,k}(ω) < x) = P (G_{n,k}(χ_{n,k}(ω)) < x) ≥ P (χ_{n,k}(ω) < F (x)) = F (x), since {ω: χ_{n,k}(ω) < F (x)} ⊂ {ω: G_{n,k}(χ_{n,k}(ω)) < x}. The last relation holds, since in the case χ_{n,k}(ω) < F_{n,k}(x) we have χ_{n,k}(ω) = F_{n,k}(x) − h with some h > 0, and G_{n,k}(χ_{n,k}(ω)) = sup{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 ≤ x − δ holds with some δ = δ(h) > 0.

Define the random variables S_{N(n,k)}(ω) = ∑_{\begin{matrix}(m,j) : m<n−1, 1≤j≤l_{m} \\ or m=n−1, and j≤k \end{matrix}} Ā_{m,j}S_{m,j}(ω), for

all n = 1, 2, …, 1 ≤ k ≤ l_{n}. If we consider the partial sums S′_{n}(ω) = ∑_{k=1}^{n} ξ_{k}(ω)

with the random variables ξ_{k}(ω) given in the formulation of Theorem 1 then the joint distribution of the random vectors S_{N(n,k)}(ω) and S′_{N(n,k)}(ω), n = 1, 2, …, 1 ≤ k ≤ 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)}(ω), p = 1, 2, …, 0 ≤ k ≤ l_{p}, can be extended to a sequence of random variables S_{n}(ω), n = 1, 2, … whose elements for the indices of the form N(p, k) are the already constructed random variables S_{N(p,k)}(ω), and the distribution of the sequences S_{n}(ω) and S′_{n}(ω), n = 1, 2, …, agree. Define the random variables ξ_{n}(ω) = S_{n}(ω) − S_{n−1}(ω), n = 1, 2, . . .. This sequence has the same joint distribution as the original sequence of independent random variables ξ_{n}(ω) considered in the formulation of Theorem 1. In such a way we constructed a sequence of random variables ξ_{n}(ω) in dependence of a small parameter ε̄ > 0.

Put ζ_{n}(ω) = ξ_{n}(ω) − η_{n}(ω), n = 1, 2, . . .. We claim that if the parameter ε̄ = ε̄(ε) is chosen sufficiently small, then this sequence together with the already constructed “ε̄ regular refinement” N(n, k) of the sequence N(n) and with the choice of the random variables ζ_{n}^{(1)}(ω) = ξ_{n}(ω), ζ_{n}^{(2)}(ω) = η_{n}(ω), ζ_{n,k,m}^{(1)}(ω) = ξ_{N(n,k−1)+m}(ω) and

ζ_{n,k,m}^{(2)}(ω) = η_{N(n,k−1)+m}(ω), n = 1, 2, …, k = 1, …, l_{n}, 0 ≤ m < N(n, k)−N(n, k −1), satisfy the Basic Lemma with parameter ε. 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 δ > 0 there is a threshold n_{0} = n_{0}(δ) such that

E(S_{n,k}(ω) − T_{n,k}(ω))^{2} ≤ δ, for all n ≥ n_{0} and 1 ≤ k ≤ l_{n}.  (4.2)

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}(ω). This result yields that the distribution functions F_{n,k} of the random variables S_{n,k}(ω) satisfy the relation lim_{n→∞} sup_{1≤k≤l_{n}} sup_{|x|<∞} |F_{n,k}(x) − Φ(x)| = 0.

This implies that the random variables T_{n,k}(ω) and the S_{n,k}(ω) constructed by the above described quantile transform from the random variables χ_{n,k}(ω) = Φ(T_{n,k}(ω)) satisfy the following relation: For all L > 1 and η > 0 there exists some n_{0} = n_{0}(L, η) such that

|S_{n,k}(ω) − T_{n,k}(ω)| < \frac{η}{L} on the set \{ω: |T_{n,k}(ω)| ≤ L\}

for all n ≥ n_{0} and 1 ≤ k ≤ l_{n}. Let us choose the number L > 1 in such a way that the standard normal random variables T_{n,k}(ω) satisfy the inequality ET_{n,k}^{2}(ω)I(|T_{n,k}(ω)| ≥

L) < \frac{δ}{10}, and let η = \frac{δ}{20}. Then E(T_{n,k}(ω) − S_{n,k}(ω))^{2}I(|T_{n,k}(ω)| ≤ L) < \frac{δ^{2}}{400}, and

|E(T_{n,k}^{2}(ω) − S_{n,k}^{2}(ω))I(|T_{n,k}(ω)| ≤ L)| ≤ (E (T_{n,k}(ω) + S_{n,k}(ω))^{2})^{1/2}

(E (T_{n,k}(ω) − S_{n,k}(ω))^{2} I(|T_{n,k}(ω)| ≤ L))^{1/2} < \frac{δ}{10}.

Since ES_{n,k}^{2}(ω) = ET_{n,k}^{2}(ω) = 1, the inequalities ES_{n,k}^{2}(ω)I(|T_{n,k}(ω)| ≤ L) ≥ 1 − \frac{δ}{5},

and ES_{n,k}^{2}(ω)I(|T_{n,k}(ω)| > L) ≤ \frac{δ}{5} hold. Because of the simple inequality

E(S_{N(n,k)}(ω) − T_{N(n,k)}(ω))^{2} ≤ E(S_{N(n,k)}(ω) − T_{N(n,k)}(ω))^{2}I(|T_{n,k}(ω)| ≤ L)

+ 2ES_{N(n,k)}^{2}(ω)I(|T_{n,k}(ω)| ≥ L) + 2ET_{N(n,k)}^{2}(ω))^{2}I(|T_{n,k}(ω)| ≥ L)

the above relations imply (4.2).

Let us remark that the estimate (4.2) does not depend on the parameter ε̄ appearing in our construction. Only the threshold index n_{0} = n_{0}(δ) = n_{0}(δ, ε̄) in (4.2) depends on this parameter.

Put

Z_{n,k}(ω) = ∑_{j=N(n,k−1)+1}^{N(n,k)} ζ_{j}(ω) = ∑_{j=N(n,k−1)+1}^{N(n,k)}(ξ_{j}(ω) − η_{j}(ω))
n = 1, 2, …, 1 ≤ k ≤ l_{n}

Observe that EZ_{n,k}(ω) = 0, EZ_{n,k}^{2}(ω) ≤ δĀ_{n,k}^{2}, if n ≥ n_{0}, ∑_{k=1}^{l_{n}} Ā_{n,k}^{2} = ∑_{j=N(n−1)+1}^{N(n)} σ_{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}(ω) = sup_{1≤j≤l_{n}} |∑_{l=1}^{j} Z_{j}(ω)|. These relations together

with the Kolmogorov inequality imply that

P (|V_{n}(ω)| ≥ εx2^{n/2}) = P ( sup_{1≤j≤l_{n}} |∑_{l=1}^{j} Z_{j}(ω)| ≥ εx2^{n/2}) ≤ \frac{δ ∑_{j=1}^{l_{n}} Ā_{n,j}^{2}}{ε^{2}x^{2}2^{n}} ≤ \frac{2δ}{ε^{2}x^{2}}

Hence we get relation (2.5) with γ = 2 by choosing δ = ε^{3}.

Now we turn to the proof of formula (2.4). We shall prove it only for the random variables ζ_{n,k,p}^{(1)}(ω) = ξ_{N(n,k−1)+p}(ω), n = 1, 2, …, k = 1, …, l_{n}, 0 ≤ p < N(n, k) − N(n, k−1). The same proof also applies to the sequence ζ_{n,k,p}^{(2)}(ω) = η_{N(n,k−1)+p}(ω), but this case can be checked simply, because the random variables ζ_{n,k,p}^{(2)}(ω) are Gaussian. The appropriate choice of the small parameter ε̄ = ε̄(ε) > 0 is important at this step.

Let us observe that by relation (4.1) the norming constants Ā_{n,k} satisfy the relation

2^{n−1}ε̄ ≤ Ā_{n,k}^{2} = B_{N(n,k)} − B_{N(n,k−1)} ≤ 2^{n+2}ε̄, and l_{n} ≤ \frac{2}{ε̄} for n ≥ n_{0}(ε̄). We have

to bound the probability of the event in formula (2.4). Since the random variables ζ_{n,k,p}^{(1)}(ω) = ξ_{N(n,k−1)+p}(ω), 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 δ > 0 and n > n_{0}(δ) there exists some K = K(δ) such that

ES_{n,k}^{2}(ω)I(|S_{n,k}(ω))| > K) ≤ 4δ.  (4.3)

Relation (4.3) follows from relation (4.2) and the observations that there exists some L > 0 such that ET_{n,k}^{2}(ω)I(T_{n,k}(ω) ≥ L) ≤ δ and some K = K(L) > 0 such that {ω: |S_{n,k}(ω)| ≥ K} ⊂ {ω: |T_{n,k}(ω)| ≥ L} for all sufficiently large n and 1 ≤ k ≤ l_{n}. The latter statement follows from the special structure of the quantile transform. Then ES_{n,k}^{2}(ω)I(|S_{n,k}(ω))| > K) ≤ 2ET_{n,k}^{2}(ω)I(|S_{n,k}(ω))| > K) + 2E(S_{n,k}(ω) − T_{n,k}(ω))^{2} ≤ 4δ, as we claimed.

We can write by the Chebishev inequality and formula (4.3) that

P (|  ∑_{p=N(n,k−1)+1}^{N(n,k)} ζ_{n,k,p}^{(1)}(ω)| > εx2^{n/2}) = P (Ā_{n,k}|S_{n,k}(ω)| > εx2^{n/2})

≤ ES_{n,k}^{2}(ω)I(|S_{n,k}(ω)| ≥ K)\frac{Ā_{n,k}^{2}}{ε^{2}x^{2}2^{n}} ≤ \frac{4Ā_{n,k}^{2}δ}{ε^{2}x^{2}2^{n}}

for all x ≥ 1, n ≥ n_{0}, and 1 ≤ k ≤ l_{n}

(4.4)

provided that ε̄ > 0 is chosen so small that Ā_{n,k}K ≤ ε2^{n/2}, which relation makes it possible to replace the second moment of S_{n,k}(ω) by the second moment of the random variable S_{n,k}(ω)I(|S_{n,k}(ω)| ≥ K) in the above estimate. Such a choice of ε̄ is possible, since Ā_{n,k}^{2} ≤ ε̄2^{n+2} if n ≥ n_{0}. (We remark that the constants in the estimations applied to get (4.4) do not depend on the parameter ε̄. Only the threshold index n_{0} depends on it.) Since l_{n}Ā_{n,k}^{2} ≤ 2^{n+3} the number δ > 0 can be chosen in such a way that δĀ_{n,k}^{2}l_{n} ≤ ε^{3}2^{n} for all sufficiently large n. For instance δ = ε^{3}/8 is a good choice. With such a choice of δ we get a weakened form of formula (2.4) with γ = 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

ξ_{1}(ω), …, ξ_{n}(ω) be independent random variables, and put S_{k}(ω) = ∑_{j=1}^{k} ξ_{j}(ω). Then

for all y > 0 such that max_{1≤k≤n} P (S_{k}(ω) > y) ≤ \frac{1}{4}

P ( sup_{1≤k≤n} S_{k}(ω) ≥ 2y) ≤ \frac{4}{3}P (S_{n}(ω) ≥ y).  (4.5)

We apply this inequality for the partial sums

S_{m}(ω) = ± ∑_{p=1}^{m} ζ_{n,k,p}^{(1)}(ω), 1 ≤ m ≤ N(n, k) − N(n, k − 1).

Since Eζ_{n,k,p}^{(1)}(ω) = 0, and the variance D_{n}^{2} = ∑_{p=N(n,k−1)+1}^{N(n,k)−N(n,k−1)} E(ζ_{n,k,p}^{(1)})^{2}(ω) = Ā_{n,k}^{2},

satisfies the inequality D_{n}^{2} = Ā_{n,k}^{2} ≤ ε̄2^{n+2}. Because of these estimates Chebishev’s inequality yields that for n ≥ n_{0}(ε̄)

max_{1≤m≤N(n,k)−N(n,k−1)} P (S_{m}(ω) > ε\frac{x}{2}2^{n/2}) ≤ \frac{1}{4} for x ≥ \frac{1}{2}

if ε̄ = ε̄(ε) is sufficiently small. Hence formula (4.5) is applicable with y = \frac{x}{2}2^{n/2} if

x ≥ \frac{1}{2}. Let us observe that relation (4.4) also holds for x ≥ 1/2 and not only for x ≥ 1. These relations imply that for sufficiently small ε̄ = ε̄(ε) > 0

P (  sup_{1≤m≤N(n,k)−N(n,k−1)} |∑_{p=1}^{m} ζ_{n,k,p}^{(1)}(ω)| > εx2^{n/2}) ≤ \frac{50Ā_{n,k}^{2}δ}{ε^{2}x^{2}2^{n}}
for all x ≥ 1, n ≥ 0, and 1 ≤ k ≤ l_{n}.

This relation implies formula (2.4) with γ = 2 in the same way as formula (4.4) implied its weakened form. Theorem 1 is proved.

Proof of Theorem 2. We may assume that lim_{x→∞} μ(x) = Eξ_{1}^{2}(ω) = ∞, because in the

case Eξ_{1}^{2}(ω) < ∞ Theorem 1 can be applied. More precisely Theorem 1 supplies a modified version of Theorem 2 in this case with the norming sequence B̄_{n} = nEξ_{1}^{2}(ω)

instead of the original sequence B_{n}. But lim_{n→∞} \frac{B̄_{n}}{B_{n}} = 1, in this case, and Theorem 5 of

Part I. implies the result in this case.

Since the function μ(x) = Eξ_{1}^{2}(ω)I(|ξ_{1}(ω)| ≤ x) is a slowly varying function, it follows for instance from Theorem 2 of [13] in Chapter VIII, Section 9 (and its proof) that

\begin{aligned}P (|ξ_{1}(ω)| > x) & = o (x^{−2}μ(x)) if x → ∞, \\ E|ξ_{1}|I(|ξ_{1}(ω)| > x) & = o (x^{−1}μ(x)) if x → ∞. \end{aligned}  (4.6)

Define the random variables ξ̄_{n}(ω) = ξ_{n}(ω)I(|ξ_{n}(ω)| ≤ a_{n}), χ_{n}^{(1)}(ω) = ξ_{n}(ω)I(ξ_{n}(ω) ≥ a_{n}), and χ_{n}^{(2)}(ω) = ξ_{n}(ω)I(ξ_{n}(ω) ≤ −a_{n}), n = 1, 2, …, with the numbers a_{n} defined in the formulation of Theorem 2. We claim that

a.) The sequence of random variables ξ̄_{n}(ω), n = 1, 2, …, satisfies the almost sure func-

tional limit theorem with the weight function B_{n} = ∑_{k=1}^{n} a_{k}, n = 1, 2, …, parameter

α = 2, and the Wiener measure μ_{0} as the limit measure. b.) Both sequences of random variables χ_{n}^{(i)}(ω), n = 1, 2, …, i = 1, 2, satisfy the almost

sure functional limit theorem with the same weight function B_{n} as in Statement a.), n = 1, 2, …, parameter α = 2, and (degenerated) limit measure μ_{0} on the space D([0, 1]) which is concentrated on the function x(t) ≡ 0.

Statement b.) can be reformulated in the following way: Put S_{n}^{(i)}(ω) = ∑_{k=1}^{n} χ_{k}^{(i)}(ω),

i = 1, 2, n = 1, 2, …, and define the broken lines S_{N}^{(i)}(·, ω), i = 1, 2, N = 1, 2, …, with the above random variables S_{n}^{(i)}(ω), the numbers B_{n} and parameter α = 2 by formula (1.2). For all ω ∈ Ω, i = 1, 2, and n = 1, 2, …, introduce the (random) measure P_{ω,i,n}

by the formula P_{ω,i,n} (S_{k}^{(i)}(·, ω)) = \frac{1}{log \frac{B_{n}}{B_{1}}} log \frac{B_{k+1}}{B_{k}}, 1 ≤ k < n. Then for any open

neighbourhhood G of the function x(t) ≡ 0 in the space D([0, 1]) lim_{n→∞} P_{ω,i,n}(G) = 0 for almost all ω ∈ Ω, i = 1, 2.

Since ξ_{n}(ω) = ξ̄_{n}(ω) + χ_{n}^{(1)}(ω) + χ_{n}^{(2)}(ω), 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 ω ∈ Ω the sequence ξ_{n}(ω), n = 1, 2, …, satisfies the same almost sure functional limit theorem as the sequence ξ̄_{n}(ω), n = 1, 2, . . ..

We shall prove Statement a.) with the help of Theorem 1. Let us first observe that

lim_{n→∞} \frac{Var ξ̄_{n}(ω)}{μ(a_{n})} = 1. Indeed, \frac{Var ξ̄_{n}(ω)}{μ(a_{n})} − 1 = \frac{(Eξ̄_{n}(ω))^{2}}{μ(a_{n})} , and by the relations (4.6)

and Eξ_{n}(ω) = 0, the definition of the sequence of a_{n} and a_{n} → ∞ as n → ∞ we have

(Eξ̄_{n}(ω))^{2} = (Eξ_{n}(ω)I(|ξ_{n}(ω)| > a_{n})^{2} = o(a_{n}^{−2}μ(a_{n})^{2}) = o (\frac{μ(a_{n})}{n}). This means that

for B̄_{n} = ∑_{k=1}^{n} Var ξ̄_{k}(ω), lim_{n→∞} \frac{B̄_{n}}{B_{n}} = 1.

To prove Statement a.) first we show that for all ε > 0

lim_{n→∞} \frac{1}{B_{n}} ∑_{k=1}^{n} E [(ξ̄_{k}(ω) − Eξ̄_{k}(ω))^{2}I (|ξ̄_{k}(ω) − Eξ̄_{k}(ω)| > εB_{n}^{1/2})] = 0,  (4.7)

which implies that the Lindeberg condition holds for the sequence ξ̄_{k}(ω) − Eξ̄_{k}(ω), k = 1, 2, . . ..

First we show that

\sqrt{c} ≤ lim inf_{n→∞} sup_{cn≤k≤n} \frac{a_{k}}{a_{n}} ≤ lim sup_{n→∞} sup_{cn≤k≤n} \frac{a_{k}}{a_{n}} ≤ 1 for all 0 < c < 1.  (4.8)

Indeed, it follows from the definition of the sequence a_{n} that a_{k} ≤ a_{n} for all k ≤ n, and this implies the right-hand side of (4.8). On the other hand, since μ(·) is a slowly varying

function the numbers a_{n} satisfy the relation lim_{n→∞} n\frac{μ(a_{n})}{a_{n}^{2}} = 1, and for any ε > 0, k ≥ cn

and n ≥ n_{0}(c, ε), k \frac{μ((\sqrt{c} − ε)a_{n})}{((\sqrt{c} − ε)a_{n})^{2}} > cn\frac{μ(a_{n})}{(c − \sqrt{c}ε)a_{n}^{2}} ≥ 1, hence a_{k} ≥ (\sqrt{c} − ε)a_{n}. This

relation implies the left-hand side of (4.8). Since μ(·) is a slowly varying function, it follows from relation (4.8) that

lim_{n→∞} \frac{B_{n}}{nμ(a_{n})} = 1.  (4.9)

By relation (4.9) the expression B_{n} can be replaced by nμ(a_{n}) in (4.7). Let us also

observe that B_{n}^{1/2} ∼ \sqrt{nμ(a_{n})} = \sqrt{\frac{nμ(a_{n})}{a_{n}^{2}}}a_{n} ∼ a_{n}, and |Eξ̄_{k}(ω)| ≤ E|ξ̄_{1}(ω)| ≤ const.

for large n. Hence the terms in the sum (4.7) can be estimated as

E [(ξ̄_{k}(ω) − Eξ̄_{k}(ω))^{2}I (|ξ̄_{k}(ω) − Eξ̄_{k}(ω)| ≥ εB_{n}^{1/2})]

≤ 2 (E|ξ_{1}(ω)|)^{2} + 2Eξ_{k}(ω)^{2}I (\frac{ε}{2}a_{n} ≤ |ξ_{k}| ≤ a_{k})

≤ const. + max (μ(a_{k}) − μ (\frac{ε}{2}a_{n}) , 0) .

Since μ(x) is a slowly varying function tending to infinity as x → ∞, lim_{n→∞} a_{n} = ∞, and

a_{k} ≥ const. \sqrt{c}a_{n} if k ≥ cn, with some 0 < c ≤ 1 the above estimate implies that for all ε > 0 and δ > 0 there is some threshold n_{0} = n_{0}(ε, δ) such that for all n ≥ n_{0} and 1 ≤ k ≤ n

E [(ξ̄_{k}(ω) − Eξ̄_{k}(ω))^{2}I (|ξ̄_{k}(ω) − Eξ̄_{k}(ω)| ≥ εB_{n}^{1/2})] ≤ δμ(a_{n}).

Summing these inequalities for all 1 ≤ k ≤ n and exploiting that they hold (for sufficiently large n) for all δ > 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 B̄_{n} such that lim_{n→∞} \frac{B̄_{n}}{B_{n}} = 1) imply that the sequence of random

variables ξ̄_{n}(ω) − Eξ̄_{n}(ω), n = 1, 2, …, satisfy the almost sure functional limit theorem with parameter α = 2 and the Wiener measure μ_{0} as limit measure. Hence to finish the proof Statement a.) it is enough to show that

lim_{n→∞} \frac{1}{B_{n}^{1/2}} ∑_{k=1}^{n} Eξ̄_{k}(ω) = 0.  (4.10)

To prove relation (4.10) let us first observe that because of the identity Eξ_{1}(ω) = 0, relations (4.6) and (4.9) we can write, fixing an ε > 0 for all r > r_{0}(ε) that

| ∑_{j=2^{r}+1}^{2^{r+1}} Eξ̄_{j}(ω)| ≤ ε ∑_{j=2^{r}+1}^{2^{r+1}} \frac{μ(a_{j})}{a_{j}} ≤ const. ε2^{r} \frac{μ(a_{2^{r}})}{a_{2^{r}}} ≤ const. εa_{2^{r}}.

Given an integer n, choose the integer R such that 2^{R} < n ≤ 2^{R+1}, and apply the above estimate for all r_{0}(ε) ≤ r ≤ R. Then we get by using the relations B_{n}^{1/2} ∼ \sqrt{nμ(a_{n})} ∼

a_{n} and a_{n} → ∞ if n → ∞ that lim_{r→∞} \frac{a_{2^{r+1}}}{a_{2^{r}}} = \sqrt{2} and

|∑_{j=1}^{n} Eξ̄_{j}(ω)| ≤ const. (ε) + const. ε ∑_{r=1}^{R} a_{2^{r}}
≤ const. (ε) + const.′εa_{n} ≤ const. (ε) + const.′εB_{n}^{1/2}.

Since the above relation holds for all ε > 0, it implies relation (4.10).

We shall prove Statement b.) for the sequence χ_{n}^{(1)}(ω), n = 1, 2, …, the proof for

the sequence χ_{n}^{(2)}(ω) is the same. Put S_{n}(ω) = ∑_{j=1}^{n} χ_{j}^{(1)}(ω). First we show that to verify

Statement b.) it is enough to prove the inequality

lim sup_{n→∞} \frac{1}{n} ∑_{k=1}^{N(n)} log \frac{B_{k}}{B_{k−1}} I (\{\frac{sup_{0≤j≤k} S_{j}(ω)}{B_{k}^{1/2}} > ε\}) ≤ ε, for all ε > 0  (4.11)

with N(n) = inf{k: B_{k} ≥ 2^{n}}. One can argue for instance in the following way. Let us remark that the sequence of (degenerated) random variables η_{n}(ω) ≡ 0 satisfies the almost sure functional limit theorem with the weight function B_{n}, parameter α = 2 and limit measure μ_{0} which is concentrated to the function x(t) ≡ 0, 0 ≤ t ≤ 1. Then to prove Statement b.) it is enough to check that the pair of sequences (χ_{n}^{(1)}(ω), η_{n}(ω)), n = 1, 2, …, 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, ζ_{n}(ω) = χ_{n}^{(1)}(ω), n = 1, 2, …, 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 ε.) 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}(ω) = sup_{N(n−1)<k≤N(n)} | ∑_{j=N(n−1)+1}^{k} ζ_{j}(ω)| = ∑_{j=N(n−1)}^{N(n)} ζ_{j}(ω),  (4.12)

and we have to prove relation (2.5) with this random variable, α = 2 and appropriate γ > 0. Then the validity of relation (2.6) for all 1 > ε > 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 ε > 0 and x ≥ 1 for the sequence

ζ_{n}(ω) = χ_{n}^{(1)}(ω) by estimating the expression ∑_{j=N(n−1)+1}^{N(n)} Eζ_{j}(ω)= E | ∑_{j=N(n−1)+1}^{N(n)} ζ_{j}(ω)|.

To do this we need a good bound on the term N(n). By the relation B_{n} ∼ nμ(a_{n}) and the definition of the numbers N(n) we have N(n)μ(a_{N(n)}) ∼ 2^{n} for sufficiently large

n. The relation a_{N(n)}^{2} \frac{N(n)μ(a_{N(n)})}{a_{N(n)}^{2}} ∼ 2^{n} and the definition of the numbers a_{n} imply

that lim_{n→∞} 2^{−n/2}a_{N(n)} = 1. Hence N(n) ≤ \frac{2^{n+2}}{μ(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)

∑_{j=N(n−1)+1}^{N(n)} Eζ_{j}(ω) ≤ ε̄N(n)\frac{μ(a_{N(n)})}{a_{N(n)}} ≤ ε̄\frac{2^{n+2}}{a_{N(n)}} ≤ ε̄2^{(n+5)/2} ≤ ε^{2}2^{n/2}

if the number ε̄ is sufficiently small (and the threshold n_{0}(ε̄) is sufficiently large). Hence the Markov inequality yields that

P ( ∑_{j=N(n−1)+1}^{N(n)} ζ_{j}(ω) ≥ εx2^{n/2}) ≤ \frac{ε^{2}2^{n/2}}{εx2^{n/2}} = εx^{−1}.

It follows from this estimate that the random variables V_{n}(ω) defined in (4.12) satisfy the estimate (2.5) with γ = 1. Theorem 2 is proved.

5. The proof of Theorem 3

Proof of Theorem 3. By the results quoted in Sections 1 and 2 it is enough to show that the sequences (ξ_{n}(ω) − a_{n}, η_{n}(ω)), n = 1, 2, …, satisfy Property A with an appropriate sequence of constants a_{n} and η_{n}(ω) = L̄(n)^{1/α}η̄_{n}(ω), n = 1, 2, …, where η̄_{n}(ω) are i.i.d. random variables with the stable distribution G(x) satisfying formula (1.5), and L̄(·) 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 ξ_{n}(ω), by means of the random variables η_{n}(ω), n = 1, 2, …, to be described below.

The random variables η_{n}(ω), n = 1, 2, …, have distribution function G_{n}(x) =

G (\frac{x}{L̄(n)^{1/α}} ) 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}(η_{n}(ω)), n = 1, 2, …, 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 ξ_{n}(ω), n = 1, 2, . . .. More explicitly, we define a sequence of i.i.d. random variables with the same distribution function F (x) as the originally given sequence ξ_{n}(ω), n = 1, 2, …, by the formula ξ_{n}(ω) = F^{−1}(G_{n}(η(ω)), n = 1, 2, …, 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 (ξ_{n}(ω) − a_{n}, η_{n}(ω)), n = 1, 2, …, with

a_{n} = E (ξ_{n}(ω) − η_{n}(ω)) I (|η_{n}(ω)| < n^{1/α}L̄(n)^{1/α})

satisfies Property A. (Observe that the construction which must satisfy certain property depending on a parameter ε > 0 does not depend on the parameter ε.)

First we formulate and prove a Lemma which plays an important role in the proof.

Lemma 1. Let us consider the function b(x) = max \{u: \frac{L(u)x}{u^{α}} ≥ 1\}, x > 0, introduced

in the formulation of Theorem 3. This function b(x) is a regularly varying function with parameter 1/α, hence L̄(x) = b(x)^{α}x^{−1} is a slowly varying function. Define the distribution functions G_{n}(x) and random variables η_{n}(ω) and ξ_{n}(ω), n = 1, 2, …, in the way described above with the help of this slowly varying function L̄(·). The relation

lim_{x→∞} \frac{xL(b(x))}{b(x)^{α}} = 1 holds. For all ε̄ > 0 there exists an index n_{0} = n_{0}(ε̄) such that

\begin{aligned}|ξ_{n}(ω) − η_{n}(ω)| < ε̄|η_{n}(ω)| & if ε̄n^{1/α}L̄(n)^{1/α} < η_{n}(ω) < ε̄^{−1}n^{1/α}L̄(n)^{1/α} \\ & and n ≥ n_{0}(ε̄). \end{aligned}  (5.1)

Proof of Lemma 1. Since L(x) is a slowly varying function we have lim_{x→∞} b(x) = ∞, and

the relations lim_{x→∞} \frac{L(b(x))x}{b(x)^{α}} = 1 and lim_{x→∞} \frac{L(c^{1/α}b(x))cx}{(c^{1/α}b(x))^{α}} = 1 hold for all 1 ≤ c ≤ K,

where K > 1 is an arbitrary fixed constant. Moreover, the convergence in the second relation is uniform in the variable c as x → ∞. It can be shown with the help of these

properties that lim_{x→∞} \frac{b(cx)}{c^{1/α}b(x)} = 1, since they imply that for large x > 0 the number

c^{1/α}b(x) is a good approximation for b(cx). This relation means that the function b(x) is regularly varying with parameter 1/α. The regular varying property of the function b(x) implies that the function L̄(x) = b(x)^{α}x^{−1} is slowly varying.

We claim that for any ε̄ > 0

lim_{n→∞} sup_{\frac{ε̄}{2}n^{1/α}L̄(n)^{1/α}<x<2ε̄^{−1}n^{1/α}L̄(n)^{1/α}} x^{α} \frac{1 − F (x)}{L(x)} = C_{1}

lim_{n→∞} sup_{\frac{ε̄}{2}n^{1/α}L̄(n)^{1/α}<x<2ε̄^{−1}n^{1/α}L̄(n)^{1/α}} x^{α} \frac{F (−x)}{L(x)} = C_{2},

(5.2)

and

lim_{n→∞} sup_{\frac{ε̄}{2}n^{1/α}L̄(n)^{1/α}<x<2ε̄^{−1}n^{1/α}L̄(n)^{1/α}} x^{α} \frac{1 − G_{n}(x)}{L(x)} = C_{1}

lim_{n→∞} sup_{\frac{ε̄}{2}n^{1/α}L̄(n)^{1/α}<x<2ε̄^{−1}n^{1/α}L̄(n)^{1/α}} x^{α} \frac{G_{n}(−x)}{L(x)} = C_{2}.

(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

lim_{n→∞} sup_{\frac{ε̄}{2}n^{1/α}L̄(n)^{1/α}<x<2ε̄^{−1}n^{1/α}L̄(n)^{1/α}} \frac{L̄(n)}{L(x)} = 1.

To see the last statement observe that in the domain we are interested in L(x) ∼ L(n^{1/α}L̄(n)^{1/α})), hence it can be reduced to the formula

lim_{n→∞} \frac{L̄(n)}{L(n^{1/α}L̄(n)^{1/α})} = 1.  (5.3)

Relation (5.3) holds, since L(n^{1/α}L̄(n)^{1/α}) = L(b(n)) ∼ \frac{b(n)^{α}}{n} = L̄(n).

Relations (5.2) and (5.2′) imply that for all ε̄ > 0 there is a threshold n = n(ε̄) such that

1 − F ((1 + ε̄)x) < 1 − G_{n}(x) < 1 − F ((1 − ε̄)x)
F (−(1 + ε̄)x) < G_{n}(−x) < F ((1 − ε̄)x)
if ε̄n^{1/α}L(n)^{1/α} < x < ε̄^{−1}n^{1/α}L(n)^{1/α}.

Hence

(1 − ε̄)x < F^{−1}(G_{n}(x)) < (1 + ε̄)x
−(1 + ε̄)x < F^{−1}(G_{n}(−x)) < −(1 − ε̄)x

if ε̄n^{1/α}L(n)^{1/α} < x < ε̄^{−1}n^{1/α}L(n)^{1/α}. The last formula together with the definition of the quantile transform imply relation (5.1). Lemma 1 is proved.

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, …, for which the numbers N(n, k) contain all integers in the interval [N(n−1), N(n)] for all n = 1, 2, . . .. 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

P (  sup_{N(n−1)<k≤N(n)} | ∑_{j=N(n−1)+1}^{k}(ξ_{j}(ω) − a_{j} − η_{j}(ω))| > εx2^{n/α}) < Cεx^{−γ}
for all x ≥ 1 and n ≥ n_{0}(ε)

(5.4)

with an appropriate γ > 0.

Let us make the following observation. Since L̄(n) is a slowly varying function the

relation lim_{n→∞} \frac{B̄_{n}}{B_{n}} = 1 holds with B̄_{n} = nL̄(n). This relation together with formula (1.1)

imply that for all large n the number N(n) satisfies the relation N(n)L̄(N(n)) ∼ 2^{n}. Since L̄(n) is a slowly varying function this relation also implies that N(n) ≤ 5N(n−1) if

n is sufficiently large. Indeed, since \frac{L̄(N(n − 1))}{L̄(N(n))} ≤ 1.01 (\frac{N(n)}{N(n − 1)})^{1/2} for sufficiently

large n, we have \sqrt{5} ≥ \frac{1.01N(n)L̄(N(n))}{N(n − 1)L̄(N(n − 1))} ≥ (\frac{N(n)}{N(n − 1)})^{1/2} for sufficiently large

n, as we claimed. Hence L̄(k) ∼ L̄(N(n)) and (kL̄(k))^{1/α} ≤ \frac{9}{8}2^{n/α} for N(n − 1) ≤ k ≤ N(n).

In the proof of the inequality (5.4) we shall bound separately the contribution of those indices j for which η_{j}(ω) is relatively small, more explicitly their values is much less than j^{1/α}L̄(j)^{1/α}, the indices j of middle order terms for which η_{j}(ω) is of order j^{1/α}L̄(j)^{1/α} and finally those indices for which the terms η_{j}(ω) are much larger than j^{1/α}L̄(j)^{1/α}. 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 ε̄ = ε̄(ε) and apply the estimate (5.1).

To estimate the contribution of the small terms let us first bound the terms

E(ξ_{j}(ω) − η_{j}(ω))I (|η_{j}(ω)| < ε^{4/(2−α)}j^{1/α}L(j)^{1/α}) − a_{j}

and

E(ξ_{j}(ω) − η_{j}(ω))^{2}I (|η_{j}(ω)| < ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α}) .

Fix a sufficiently small ε̄ = ε̄(ε) to be defined later, and consider indices j > n_{0}(ε̄). i.e. such indices for which the relation (5.1) is applicable. In this case, if ε̄(ε) is sufficiently small, we can write because of the definition of the norming constants a_{n}

\begin{matrix}|E(ξ_{j}(ω) − η_{j}(ω))I (|η_{j}(ω)| < ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α}) − a_{j}| \\ = |E(ξ_{j}(ω) − η_{j}(ω))I (ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α} ≤ |η_{j}(ω)| < j^{1/α}L̄(j)^{1/α})| \\ ≤ ε̄E|η_{j}(ω)|I (ε^{4/(2−α)}j^{1/α}L(j)^{1/α} ≤ |η_{j}(ω)| < j^{1/α}L̄(j)^{1/α}) \\ = ε̄∫_{ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α}≤|u|≤j^{1/α}L̄(j)^{1/α}} |u|G_{j}( du) \\ ≤ const. ε̄L̄(j)^{1/α} ∫_{ε^{4/(2−α)}j^{1/α}}^{j^{1/α}} uḠ( du) ≤ \frac{ε}{45}j^{(1−α)/α}L̄(j)^{1/α} \end{matrix} (5.5)

with Ḡ(u) = 1 − G(u) + G(−u). To estimate the second moment of these summands write

E(ξ_{j}(ω) − η_{j}(ω))^{2}I (|η_{j}(ω)| < ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α})
≤ 2Eξ_{j}^{2}(ω)I (|ξ_{j}(ω)| < 2ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α})
+ 2Eη_{j}^{2}(ω)I (|η_{j}(ω)| < ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α}) .

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 η = 0 and ζ = 2, that

E(ξ_{j}(ω) − η_{j}(ω))^{2}I (|η_{j}(ω)| < ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α})

≤ const. ε^{8/(2−α)}j^{2/α}L̄(j)^{2/α}[(1 − F (ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α}))
+ (1 − G_{j}(ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α}))]

≤ const. ε^{8/(2−α)}j^{2/α}L̄(j)^{2/α}ε^{−4α/(2−α)}j^{−1}L̄(j)^{−1} [L (j^{1/α}L̄(j)^{1/α}) + L̄(j)]

≤ const. ε^{4}j^{−1+2/α}L̄(j)^{2/α}

(5.6)

because of formula (5.3).

The estimates (5.5) and (5.6) together with the estimates N(n)L̄(N(n)) ∼ 2^{n}, (kL̄(k))^{1/α} ≤ \frac{9}{8}2^{n/α} for k ≤ N(n), N(n) < 5N(n − 1) and Kolmogorov’s inequality imply that with the notation of the events A_{j} = {ω: |η_{j}(ω)| < ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α}} and their indicator functions I(A_{j}(ω)), j = 1, 2, …,

| ∑_{j=N(n−1)+1}^{k} E(ξ_{j}(ω) − η_{j}(ω))I(A_{j}(ω)) − a_{j})| ≤ ∑_{j=N(n−1)+1}^{k} \frac{ε}{45}j^{(1−α)/α}L̄(j)^{1/α}

≤ \frac{ε}{40}2^{n/α} ∑_{j=N(n−1)+1}^{N(n)} \frac{1}{j} ≤ \frac{ε}{30}2^{n/α} log \frac{N(n)}{N(n − 1)} ≤ \frac{ε}{8}2^{n/α}

for all N(n − 1) < k ≤ N(n), and

P ( sup_{N(n−1)<k≤N(n)} | ∑_{j=N(n−1)+1}^{k}(ξ_{j}(ω) − η_{j}(ω))I(A_{j}(ω)) − a_{j}| > \frac{ε}{4}x2^{n/α})

≤ \frac{∑_{j=N(n−1)+1}^{N(n)} E(ξ_{j}(ω) − η_{j}(ω))^{2}I(A_{j}(ω))}{\frac{ε^{2}}{64}x^{2}2^{2n/α}} ≤ \frac{ε^{4} [N(n)L̄(N(n))]^{2/α}}{\frac{ε^{2}}{64}x^{2}2^{2n/α}}

≤ \frac{const. ε^{2}}{x^{2}} for all x ≥ 1 and n ≥ n_{0}(ε̄)

(5.7)

We still have to bound the contribution of those terms ξ_{j}(ω) − η_{j}(ω) in the sum (5.4) for which η_{j}(ω) ≥ ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α}. We shall separate them to middle terms for which |η_{j}(ω)| is not too large and large terms for which |η_{j}(ω)| 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 ε̄(ε) sufficiently small, and define the sets

B_{j}(x) = {ω: ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α} ≤ |η_{j}(ω)| < ε^{−1/α}xj^{1/α}L̄(j)^{1/α}}

and

C_{j}(x) = {ω: |η_{j}(ω)| ≥ ε^{−1/α}xj^{1/α}L̄(j)^{1/α}}

together with the indicator functions I(B_{j}(x)(ω)) and I(C_{j}(x)(ω)), j = 1, 2, …, of these sets if 1 ≤ x ≤ ε̄^{−1/4}. For sufficiently large j relation (5.1) can be applied on the sets B_{j}(ω). This implies that because of the factor ε̄ in the upper bound given (5.1) we can prove similarly to the proof of relations (5.5) and (5.6) that

|E(ξ_{j}(ω) − η_{j}(ω))I(B_{j}(x)(ω))| ≤ \frac{ε}{45}j^{(1−α)/α}L̄(j)^{1/α}.  (5.8)

and

E(ξ_{j}(ω) − η_{j}(ω))^{2}I(B_{j}(x)(ω)) ≤ const. ε^{4}j^{−1+2/α}L̄(j)^{2/α}  (5.9)

for x ≤ ε̄^{−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

| ∑_{j=N(n−1)+1}^{k} E(ξ_{j}(ω) − η_{j}(ω))I(B_{j}(ω))| ≤ ∑_{j=N(n−1)+1}^{k} \frac{ε}{45}j^{(1−α)/α}L̄(j)^{1/α}

≤ \frac{ε}{9} (N(n)L̄(N(n)))^{1/α} ≤ \frac{ε}{8}2^{n/α}

for all N(n−) + 1 ≤ k ≤ N(n) and

P ( sup_{N(n−1)<k≤N(n)} | ∑_{j=N(n−1)+1}^{k}(ξ_{j}(ω) − η_{j}(ω))I(B_{j}(x)(ω))| > \frac{ε}{4}x2^{n/α})

≤ \frac{∑_{j=N(n−1)+1}^{N(n)} E(ξ_{j}(ω) − η_{j}(ω))^{2}I(B_{j}(x)(ω))}{\frac{ε^{2}}{64}x^{2}2^{2n/α}} ≤ const. \frac{ε^{4} [N(n)L̄(N(n))]^{2/α}}{\frac{ε^{2}}{64}x^{2}2^{2n/α}}

≤ \frac{const. ε^{2}}{x^{2}} for all ε̄^{−1/4} ≥ x ≥ 1 and n ≥ n_{0}(ε̄).

(5.10)

In the outer domain we can write for ε̄^{−1/4} ≥ x because of the definition of the sets C_{j}(x)

P ( sup_{N(n−1)<k≤N(n)} | ∑_{j=N(n−1)+1}^{k}(ξ_{j}(ω) − η_{j}(ω))I(C_{j}(x)(ω))| > \frac{ε}{2}x2^{n/α})

≤ P ({ω: ω ∈ C_{j}(x) for some N(n − 1) < k ≤ N(n)})

≤ ∑_{j=N(n−1)+1}^{N(n)} P (|η_{j}(ω)| ≥ ε^{−1/α}xj^{1/α}L̄(j)^{1/α})

≤ const. εx^{−α} ∑_{j=N(n−1)+1}^{N(n)} \frac{1}{j} ≤ const. εx^{−α}.

(5.11)

Since A_{j} ∪ B_{j}(x) ∪ C_{j}(x) = Ω for all j, relations (5.7), (5.10) and (5.11) imply relation (5.4) with γ = α in the case x < ε̄^{−1/4}.

In the case x ≥ ε̄^{−1/4} define the sets

B′_{j}(x) = {ω: ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α} ≤ |η_{j}(ω)| < x^{1/2}j^{1/α}L̄(j)^{1/α}}

and

C′_{j}(x) = {ω: |η_{j}(ω)| ≥ x^{1/2}j^{1/α}L̄(j)^{1/α}}

together with the indicator functions of these sets I(B′_{j}(x)(ω)) and I(C′_{j}(x)(ω)), j = 1, 2, . . .. We remark that if ε̄(ε) > 0 is very small, then x^{1/2} ≪ εx. In this case we can apply the estimation

L (x^{1/2}j^{1/α}L̄(j)^{1/α}) ≤ C(t)x^{t}L(j^{1/α}L̄(j)^{1/α}) ≤ C′(t)x^{t}L̄(j)  (5.12)

for all x ≥ 1 and t > 0 which holds, because L(·) 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.

P ( sup_{N(n−1)<k≤N(n)} | ∑_{j=N(n−1)+1}^{k}(ξ_{j}(ω) − η_{j}(ω))I(C′_{j}(x)(ω))| > \frac{ε}{2}x2^{n/α})

≤ ∑_{j=N(n−1)+1}^{N(n)+1} P (|η_{j}(ω)| ≥ x^{1/2}j^{1/α}L̄(j)^{1/α})

≤ const. x^{−α/3} ∑_{j=N(n−1)+1}^{N(n)} \frac{1}{j} ≤ const. x^{−α/3} < εx^{−α/4},

(5.13) if ε̄ 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 (ξ_{j}(ω) − η_{j}(ω))I(B′_{j}(x))(ω).

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 η = 0 and ζ = 2. Then we get with the application of formula (5.12) that

E(ξ_{j}(ω) − η_{j}(ω))^{2}I(B′_{j}(x)(ω)) ≤ const. xj^{2/α}L̄(j)^{2/α}
[(1 − F (x^{1/2}j^{1/α}L̄(j)^{1/α})) + (1 − G_{j}(x^{1/2}j^{1/α}L̄(j)^{1/α}))]
≤ const. xj^{2/α}L̄(j)^{2/α}x^{−α/2}j^{−1}L̄(j)^{−1} [L (x^{1/2}j^{1/α}L̄(j)^{1/α}) + L̄(j)]
≤ const. xj^{−1+2/α}L̄(j)^{2/α},

and

Var ( ∑_{j=N(n−1)+1}^{N(n)}(ξ_{j}(ω) − η_{j}(ω))I(B′_{j}(x))(ω)) ≤ const. x2^{2n/α}.  (5.14)

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 ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α} ≤ |u| ≤ j^{1/α}L̄(j)^{1/α}. Hence, handling this part of the integral separately we get integrating by parts and applying (5.12) with t = α > 0

|E(ξ_{j}(ω) − η_{j}(ω))I(B′_{j}(x)(ω))| ≤ εj^{−1+1/α}L̄(j)^{1/α}
+ ∫_{j^{1/α}L̄(j)^{1/α}≤|u|≤x^{1/2}j^{1/α}L̄(j)^{1/α}} |u|(F ( du) + G_{j}(du))
≤ const. ∫_{j^{1/α}L̄(j)^{1/α}≤|u|≤x^{1/2}j^{1/α}L̄(j)^{1/α}}(F (u) + G_{j}(u)) du
+ const. x^{1/2}j^{−1+1/α}L̄(j)^{1/α} ≤ const. x^{1/2}j^{−1+1/α}L̄(j)^{1/α}.

Hence

|E ∑_{j=N(n−1)+1}^{k}(ξ_{j}(ω) − η_{j}(ω))I(B′_{j}(x)(ω))|

≤ const. x^{1/2}N(n)^{1/α}L̄(N(n))^{1/α} ≤ const. x^{1/2}2^{n/α} ≤ \frac{ε}{8}x2^{n/α}

(5.15)

for all N(n−1) ≤ k ≤ N(n) if ε̄ = ε̄(ε) is sufficiently small, hence x > ε̄^{−1/4} is sufficiently large. The estimates (5.14) and (5.15) together with Kolmogorov’s inequality imply that

P ( sup_{N(n−1)<k≤N(n)} | ∑_{j=N(n−1)+1}^{k}(ξ_{j}(ω) − η_{j}(ω))I(B′_{j}(x)(ω))| > \frac{ε}{4}x2^{n/α})

≤ \frac{x2^{2n/α}}{\frac{ε^{2}}{64}x^{2}2^{2n/α}} ≤ \frac{ε}{x^{1/2}} for all x ≥ ε̄^{−1/4} and n ≥ n_{0}(ε̄)

(5.16)

if ε̄ is sufficiently small. Now relations (5.7), (5.13) and (5.16) imply relation (5.4) in remaining case x > ε̄^{−1/4} with γ = min (\frac{α}{4}, \frac{1}{2}) = \frac{α}{4}. Since the number ε̄ = ε̄(ε) 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 vari-

ables \frac{1}{B_{n}} ∑_{j=1}^{n}(ξ_{j}(ω) − a_{n}) converge in distribution to the distribution function G(x).

We prove this if we show that \frac{1}{n^{1/α}L̄(n)^{1/α}} ∑_{j=1}^{n}(ξ_{j}(ω) − η_{j}(ω) − a_{n}) ⇒ 0 as n → ∞,

where ⇒ denotes convergence in distribution. (The denominator B_{n} can be replaced by n^{1/α}L̄(n)^{1/α} ∼ B_{n} in this formula.) Let us choose the representation

ξ_{j}(ω) − η_{j}(ω) = (ξ_{j}(ω)) − η_{j}(ω))I (|η_{j}(ω)| < ε^{4/(2−α)}n^{1/α}L̄(n)^{1/α})

+ (ξ_{j}(ω)) − η_{j}(ω))I (ε^{4/(2−α)}n^{1/α}L̄(n)^{1/α}) ≤ |η_{j}(ω)| ≤ ε^{−1}n^{1/α}L̄(n)^{1/α})

+ (ξ_{j}(ω)) − η_{j}(ω))I (|η_{j}(ω)| ≥ ε^{−1}n^{1/α}L̄(n)^{1/α}) ,

(5.17)

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 ε^{4/(2−α)}n^{1/α}L̄(n)^{1/α} instead of ε^{4/(2−α)}j^{1/α}L̄(j)^{1/α} in relations (5.5) and (5.6), and ε^{−1}n^{1/α}L̄(n)^{1/α} instead of xj^{1/α}L̄(j)^{1/α} 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 ≤ j ≤ 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, …, n by means of Chebishev’s inequality. We get that

P (\frac{1}{n^{1/α}L̄(n)^{1/α}} ∑_{j=1}^{n} ((ξ_{j}(ω) − η_{j}(ω))I(|η_{j}(ω)| < ε^{−1}n^{1/α}L̄(n)^{1/α}) − a_{n}) > ε)
≤ const. ε^{2}.  (5.18)

On the other hand,

∑_{j=1}^{n} P (|η_{j}(ω)| ≥ ε^{−1}n^{1/α}L̄(n)^{1/α}) ≤ n × const. ε^{α}n^{−1} = const. ε^{α}  (5.19)

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.)

Proof of Theorem 3′. The stable process X_{0}(t, ω) is not self-similar, because only the relation

X_{0}(tT, ω) =^{Δ} T X_{0}(t, ω) + γtT log T

holds with γ = C_{1} − C_{2}, where =^{Δ} denotes equation in distribution. In this case the process X′_{0}(t, ω) = X_{0}(t, ω) − γt log t is self-similar with self-similarity parameter α = 1. Indeed, the relation

X′_{0}(tT, ω) = X_{0}(tT, ω) − γtT log tT =^{Δ} T X_{0}(t, ω) + γtT (log T − log tT)
= T (X_{0}(t, ω) − γt log t) = T X′_{0}(t, ω)

holds for all T > 0. The results of Part I. can be applied for this process. They imply that the random variables η_{n}(ω) = L̄(n)η̄_{n}(ω) − C_{n} satisfy the almost sure functional limit theorem with C_{n} = γ(B_{n} log B_{n} − B_{n−1} log B_{n−1}), where η_{j}(ω), n = 1, 2, …, 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 α = 1. They imply that relation (5.4) also holds in this case. Hence Property A holds for the pairs (ξ_{n}(ω) − C_{n}, η_{n}(ω) − C_{n}). This result implies that the sequence ξ_{n}(ω) − C_{n} or the sequence ξ_{n}(ω) with an appropriate (modified) shift a_{n} and weight function

B_{n} = ∑_{k=1}^{n} L̄(k) satisfies the almost sure functional limit theorem.

6. Discussion of the results, and some open problems

Our results state that for a sequence of random variables ξ_{n}(ω), n = 1, 2, …, with some nice properties a result of the following type holds. Define a sequence of random broken lines X_{n}(·, ω), n = 1, 2, …, in the way described in formula (1.2) by means of these random variables ξ_{1}(ω), ξ_{2}(ω), …, for all n = 1, 2…., and ω ∈ Ω, define the probability μ_{N}(ω) 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}(·, ω), k = 1, …, N, ∑_{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 μ_{N}(ω) weakly converge to an appropriate measure μ_{0} for almost all ω ∈ Ω. 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 Csáki and Földes. 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 μ_{N}(ω), N = 1, 2, …, are to the limit measure μ_{0}. The following two questions seem to be natural problems in this direction.

i.) Can a more precise estimate be given about the distance d(μ_{N}(ω), μ_{0}), where d(·, ·) is an appropriate metric on the space of probability measures which metrizes weak

convergence? With which replacement of the weights \begin{matrix}log & \frac{B_{k+1}}{B_{k}} \\ log & \frac{B_{N}}{B_{1}} \end{matrix} in formula (1.3) do

the measures μ_{N}(ω), N = 1, 2, …, have the same limit for almost all ω ∈ Ω as the original measures in the definition of the almost sure invariance principle?

ii.) The weak convergence of the measures μ_{N}(ω) to μ_{0} states that ∫ F(x)μ_{N}(ω)( dx) → ∫ F(x)μ_{0}( dx) as N → ∞ for all continuous and bounded functional F in the space D([0, 1]) (or C([0, 1]). For which larger classes of functionals F does this statement hold?

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 FS_{k}(·, ω) for almost all ω ∈ Ω, where the trajectories S_{k}(·, ω) 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 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, ω). 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}(ω) = ∑_{j=1}^{n} ξ_{j}(ω), n = 1, 2, …, of

independent random variables with expectation zero satisfy the relation

lim_{n→∞} \frac{1}{∑_{k=1}^{n} b_{k}} ∑_{k=1}^{n} \frac{1}{b_{k}} P (S_{k} < x \sqrt{Var S_{k}}) = Φ(x) for almost all ω ∈ Ω

with the choice of appropriate weights b_{n}, n = 1, 2, …, where Φ(·) denotes the normal distribution function. The result of M. Atlagh [1] is the sharpest result in this direction among the results I know about. Atlagh assumed, similarly to our Theorem 1, that the Lindeberg condition holds, and he also formulated a very weak restriction about

the growth of the variance of the summands. He chose the weights b_{k} = \frac{Eξ_{k}^{2}}{Var S_{k}} in his

paper. Formally, this is a choice of weights different from ours. But since log \frac{Var S_{k}}{Var S_{k−1}} ∼

\frac{Eξ_{k}^{2}}{Var S_{k}}, and this approximation is sufficiently good under the conditions formulated by

Atlagh, it can be proved with the help of Theorem 4 in Part I. that the results with these two different weights are equivalent. More explicitly, it follows from this result

that the two choices of weights are equivalent if ∑_{k=1}^{∞} (\frac{Eξ_{k}^{2}}{Var S_{k}} )^{2} < ∞, and this relation

holds under the conditions of Atlagh’s result. We omit the details of the proof.

We discuss the analogs of Theorems 2 and 3 more briefly. 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 ≤ k ≤ 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 α < 1 and a_{n} = Eξ_{1}(ω) in the case 1 < α < 2. The choice of the norming constant a_{n} cannot be given in such a simple way in the case α = 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.

REFERENCES

1. Atlagh, M.: Théorème central limite presque sûr et loi du logarithme itéré pour des sommes de variables aléatoires indépendantes. C. R. Acad. Sci. Paris Sér. I. 316 (1993), 929–933.

2. Berkes, I.: On the almost sure central limit theorem and domains of attraction. Probab. Theory Rel. Fields 102 (1995), 1–18.

3. Berkes. I.: Results and problems related to the pointwise central limit theorem. In Asymptotic Methods in Probability and Statistics, Elsevier, (1998) 59–96.

4. Berkes, I. and Csáki, E.: On the pointwise central limit theorem and mixtures of stable distributions. Stat. Prob. Letters 29 (1996), 361–368.

5. Berkes, I. and Dehling, H.: Some limit theorems in log density. Ann. Probab. 21 (1993), 1640–1670.

6. Berkes, I. and Dehling, H.: On the almost sure central limit theorem for random variables with infinite variance. J. Theor. Probab. 7 (1994), 667–680.

7. Bingham, N. H. and Rogers, L. C. G.: Summability methods and almost-sure convergence, in: A. Bellow and R. Jones, eds., Almost everywhere convergence II. Academic Press, New York, 1991, pp. 69–83.

8. Breiman, L.: Probability, Addison–Wesley Publishing Company, Reading Massachusetts, Menlo Park California, London, Don Mills Ontario (1968)

9. Csáki, E. and Földes, A.: On two ergodic properties of self-similar processes. In Asymptotic Methods in Probability and Statistics, Elsevier, (1998) 97–111.

10. Dobrushin, R. L.: Gaussian and their subordinated self-similar random fields. Annals of Probability 7 No. 1. (1979) 1–28.

11. Dobrushin, R. L. and Major, P.: Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheorie 50 (1979) 27–52.

12. Dudley, R. M.: A course on empirical processes. Ecole d’Eté de Probabilités de Saint Flour XII. 1982 Lecture Notes in Mathematics, 1093 1–142, Springer, New York

13. Feller, W.: An introduction to probability theory and its applications, Vol. II. Wiley & Sons Inc. New-York–Sidney–Toronto, 1971.

14. Horváth, L. and Khoshnevisan, D.: A strong approximation for logarithmic averages. Studia Sci. Math. Hung. 31 (1996), 187–196.

15. Ibragimov, I. A. and Lifshitz, M. A.: O predel’nih teoremah tipa “pochti navernoe” (On “almost sure” type limit theorems), In Russian, Teoriya Veroyatnostej, (to appear)

16. Ibragimov, I. A. and Lifshitz, M. A.: On the convergence of generalized moments in the almost sure central limit theorem (to appear)

17. Lamperti, J. W.: Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962) 64–78.

18. Major, P.: Almost sure functional limit theorems. Part I. The general case. Studia Sci. Math. Hung. 34 (1998), 273–304.

19. Taqqu, M. S.: Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheorie 50 (1979) 53–83.