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Generalizing results of Schatte [11] and Atlagh and Weber [2], in this paper we give conditions for a sequence of random variables to satisfy the almost sure central limit theorem along a given sequence of integers.

## Abstract

*m*≥ 1. Using some classical bounds of character sums, we estimate the average value of the character sums with subsequence sums

*N*-element sequences

*S*= (

*s*

_{1}, …,

*s*

*N*) of integer elements in a given interval [

*K*+ 1,

*K*+

*L*]. In particular, we show that

*T*

_{ m }(

*S*, χ) is small on average over all such sequences. We apply it to estimating the number of perfect squares in subsequence sums in almost all sequences.

## Abstract

The concepts of subsequence and rearrangement of double sequence are used to present multidimensional analogues of the following
core questions. If *x* is a bounded real sequence and *A* is a matrix summability method, under what conditions does there exist *y*, a subsequence (rearrangement) of *x* such that each number *t* in the core of *x* is a limit point of *Ay*?

Summary For a finite abelian group *G*, we investigate the invariant s(*G*) (resp. the invariant s_{0}(*G*)) which is defined as the smallest integer *l* ? **N** such that every sequence *S *in *G* of length |*S*| = *l* has a subsequence *T* with sum zero and length |*T*|= exp(*G*) (resp. length |*T*|=0 mod exp(*G*)).

## Abstract

*s*

_{ n }are studied under the assumption that a subsequence

*η*

_{ k }} is chosen, by means of which a majorant of the terms

*u*

_{ n }is constructed. Conditions on {

*n*

_{ k }} and {

*η*

_{ k }} are found which imply the (

*C*, 1)-summability of the series

*∑ u*

_{ n }(Theorem 1). In the meanwhile, it is proved that the (

*C*, 1)-means in Theorem 1 cannot be replaced by (

*C*, α)-means, if 0<α<1 (Theorem 2). On the other hand, if the assumption in Theorem 1 is not satisfied, then in certain cases the series

*∑ u*

_{ n }preserves the property of (

*C*, 1)-summability (Theorems 4 and 5), while in other cases it is not summable even by Abel means (Theorems 3 and 6).

Mathematical Society . [3] Chi , R. , Ding , S. , Gao , W. , Geroldinger , A. , Schmid , W. A. 2005 On zero-sum subsequences of restricted size. IV Acta Math. Hungar. 107 337 – 344 10