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## On the Almost Sure Central Limit Theorem Along Subsequences

Mathematica Pannonica
Authors:
István Berkes
and
Endre Csáki

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.

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## Character sums with subsequence sums

Periodica Mathematica Hungarica
Authors:
Sanka Balasuriya
and
Igor Shparlinski

## Abstract

Let χ be a primitive multiplicative character modulo an integer m ≥ 1. Using some classical bounds of character sums, we estimate the average value of the character sums with subsequence sums
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$T_m (\mathcal{S},\chi ) = \sum\nolimits_{\mathcal{I} \subseteq \{ 1, \ldots ,N\} } {\chi (\sum\nolimits_{i \in \mathcal{I}} {s_i } )}$$ \end{document}
taken over all 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.
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## Core theorems for double subsequences and rearrangements

Acta Mathematica Hungarica
Authors:
H. Miller
and
R. Patterson

## 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?

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## On zero-sum subsequences of restricted size. IV

Acta Mathematica Hungarica
Authors:
Rui Chi
,
Shuyan Ding
,
Weidong Gao
,
Alfred Geroldinger
, and
Wolfgang A. Schmid

Summary For a finite abelian group G, we investigate the invariant  s(G) (resp.  the invariant  s0(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)).

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## Interrelations between convergence of subsequences of partial sums of numerical series and its summability by (C, α) and Abel means

Analysis Mathematica
Author:
Д. Меняшов

## Abstract

Numerical series
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop \Sigma \limits_{n = 0}^\infty u_n$$ \end{document}
with partial sumss n are studied under the assumption that a subsequence
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {S_{n_k } } \right\}_{k = 0}^\infty$$ \end{document}
of the partial sums is convergent. Then a sequence {η k } is chosen, by means of which a majorant of the termsu 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).
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## An extension of the Komlós subsequence theorem

Acta Mathematica Hungarica
Author:
I. Berkes
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## On Almost Convergent and Statistically Convergent Subsequences

Acta Mathematica Hungarica
Authors:
H. Miller
and
C. Orhan
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## Strong summability and convergence of subsequences of orthogonal series

Acta Mathematica Hungarica
Author:
H. Schwinn
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## Zero-sum problems with congruence conditions

Acta Mathematica Hungarica
Authors:
Alfred Geroldinger
,
David J. Grynkiewicz
, and
Wolfgang A. Schmid

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

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## Some examples of application of the metric entropy method

Acta Mathematica Hungarica
Author:
Mihel Weber
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