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. Probab. Lett. 79 105 – 111 10.1016/j.spl.2008.07.026 . [3] Fazekas , I. , Klesov , O. 2000 A general approach to the strong laws of large numbers Teor. Verojatnost. i Primenen. 45 569

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On the strong law of large numbers for pairwise independent random variables Acta Math. Hungar. 42 319 – 330 10.1007/BF01956779 . [3

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Abstract  

Random forests are studied. A moment inequality and a strong law of large numbers are obtained for the number of trees having a fixed number of nonroot vertices.

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Moment inqualities and strong laws of large numbers are proved for random allocations of balls into boxes. Random broken lines and random step lines are constructed using partial sums of i.i.d. random variables that are modified by random allocations. Functional limit theorems for such random processes are obtained.

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Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X 1,X 2, … is any sequence of integrable i.i.d. random variables, then

\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} $$\mathop {\lim }\limits_{N \to \infty } \frac{{\sum\nolimits_{n = 1}^N {f(n)X_n } }} {{\sum\nolimits_{n = 1}^N {f(n)} }} = \mathbb{E}Xa.s.$$ \end{document}

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References [1] Batty , C. J. K. 1979 The strong law of large numbers for states and traces of a W*-algebra Z. Wahrsch. Verw

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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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