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# Almost sure limit theorems of mantissa type for semistable domains of attraction

Acta Mathematica Hungarica
Author: P. Becker-Kern

## Abstract

A certain class of stochastic summability methods of mantissa type is introduced and its connection to almost sure limit theorems is discussed. The summability methods serve as suitable weights in almost sure limit theory, covering all relevant known examples for, e.g., normalized sums or maxima of i.i.d. random variables. In the context of semistable domains of attraction the methods lead to previously unknown versions of semistable almost sure limit theorems.

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# Weak invariance principle for mixing sequences in the domain of attraction of normal law

Studia Scientiarum Mathematicarum Hungarica
Authors: Raluca Balan and Rafał Kulik
In this article we prove a weak invariance principle for a strictly stationary φ -mixing sequence { X j } j≧1 , whose truncated variance function
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$L(x): = EX_1^2 1_{\{ |X_1 | \leqq _x \} }$$ \end{document}
is slowly varying at ∞ and mixing coefficients satisfy the logarithmic growth condition: Σ n ≧1 φ 1/2 (2 n ) < ∞. This will be done under the condition that
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim }\limits_n Var\left( {\sum\limits_{j = 1}^n {\hat X_j } } \right)/\left[ {\sum\limits_{j = 1}^n {Var (\hat X_j )} } \right] = \beta ^2$$ \end{document}
exists in (0, ∞), where
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\hat X_j = X_j I_{\{ |X_j | \leqq \eta _j \} }$$ \end{document}
and η n 2nL ( η n ).
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# A strong approximation theorem for sums of random vectors in the domain of attraction to a stable law

Acta Mathematica Hungarica
Authors: I. Berkes, A. Dabrowski, H. Dehling, and W. Philipp
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# Integral analogues of almost sure limit theorems

Periodica Mathematica Hungarica
Authors: Alexey Chuprunov and István Fazekas

Summary An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.

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# A note on asymptotics of linear combinations of iid random variables

Periodica Mathematica Hungarica
Author: Péter Kevei

## Abstract

Let X1,X2, ... be iid random variables, and let a n = (a 1,n, ..., a n,n) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination
\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} $$S_{a_n }$$ \end{document}
= a 1,n X 1 + ... + a n,n X n under the natural negligibility condition limn→∞ max{|a k,n|: k = 1, ..., n} = 0. We prove that if
\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} $$S_{a_n }$$ \end{document}
is asymptotically normal for a weight sequence a n, in which the components are of the same magnitude, then the common distribution belongs to
\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} $$\mathbb{D}$$ \end{document}
(2).
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# Marcinkiewicz laws with infinite moments

Acta Mathematica Hungarica
Author: Z. Szewczak

## Abstract

Marcinkiewicz laws of large numbers for φ-mixing strictly stationary sequences with r-th moment barely divergent, 0 < r < 2, are established. For this dependent analogs of the Lévy-Ottaviani-Etemadi and Hoffmann-Jørgensen inequalities are revisited.

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# On weighted approximations in D [0, 1] with applications to self-normalized partial sum processes

Acta Mathematica Hungarica
Authors: M. Csörgő, B. Szyszkowicz, and Q. Wang

## Abstract

Let X,X 1,X 2,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S [nt], 0 ≦ t ≦ 1} where S n = Σj=1 n X j, under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S [nt]=V n, 0 ≦ t ≦ 1}, where V n 2 = Σj=1 n X j 2. L p approximations of self-normalized partial sum processes are also discussed.

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# One-sided strong laws forincrements of sumsof i.i.d. random variables

Studia Scientiarum Mathematicarum Hungarica
Author: A. N. Frolov

We find a universal norming sequence in strong limit theorems for increments of sums of i.i.d. random variables with finite first moments and finite second moments of positive parts. Under various one-sided moment conditions our universal theorems imply the following results for sums and their increments: the strong law of large numbers, the law of the iterated logarithm, the Erdős-Rényi law of large numbers, the Shepp law, one-sided versions of the Csörgő-Révész strong approximation laws. We derive new results for random variables from domains of attraction of a normal law and asymmetric stable laws with index αЄ(1,2).

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# Asymptotics of Studentized U-type processes for changepoint problems

Acta Mathematica Hungarica
Authors: M. Csörgõ, B. Szyszkowicz, and Q. Wang

## Abstract

This paper investigates weighted approximations for Studentized U-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition E|h(X 1, X 2)|2 < ∞ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence.

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# Law of the iterated logarithm for self-normalized sums and their increments

Studia Scientiarum Mathematicarum Hungarica
Authors: Han-Ying Liang, Jong-Il Baek, and Josef Steinebach

Let X 1, X 2,… be independent, but not necessarily identically distributed random variables in the domain of attraction of a stable law with index 0<a<2. This paper uses M n=max 1 ? i ? n|X i| to establish a self-normalized law of the iterated logarithm (LIL) for partial sums. Similarly self-normalized increments of partial sums are studied as well. In particular, the results of self-normalized sums of Horváth and Shaounder independent and identically distributed random variables are extended and complemented. As applications, some corresponding results for self-normalized weighted sums of iid random variables are also concluded.

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