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Abstract  

The well-known characterization indicated in the title involves the moving maximal dyadic averages of the sequence (X k: k = 1, 2, …) of random variables in Probability Theory. In the present paper, we offer another characterization of the SLLN which does not require to form any maximum. Instead, it involves only a specially selected sequence of moving averages. The results are also extended for random fields (X k: k, ℓ = 1, 2, …).

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Abstract  

A paper by Chow [3] contains (i.a.) a strong law for delayed sums, such that the length of the edge of the nth window equals n α for 0 < α < 1. In this paper we consider the kind of intermediate case when edges grow like n=L(n), where L is slowly varying at infinity, thus at a higher rate than any power less than one, but not quite at a linear rate. The typical example one should have in mind is L(n) = log n. The main focus of the present paper is on random field versions of such strong laws.

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Пусть (X k),k=1,2,... — последов ательность случайны х переменных с нулевым средним и конечной дисперсие йσ k 2 Пусть α>1; если (X k) та кова, чтоE(X k X t)=0 дляp α<k<1≦(p+1)α (p,k,l=1, 2, ...) и, более того, такова, чт о
\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} $$\sum\limits_{k = 1}^\infty {\frac{{\sigma _k^2 }}{{k^{2 - 1/\alpha } }}(\log k)^2< + \infty }$$ \end{document}
, то 1/n (X 1+...+X n)→0 почти навер ное приn→∞.
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Abstract  

This note concerns the asymptotic behavior of a Markov process obtained from normalized products of independent and identically distributed random matrices. The weak convergence of this process is proved, as well as the law of large numbers and the central limit theorem.

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