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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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. 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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Summary General linear combinations of independent winnings in generalized \St~Petersburg games are interpreted as individual gains that result from pooling strategies of different cooperative players. A weak law of large numbers is proved for all such combinations, along with some almost sure results for the smallest and largest accumulation points, and a considerable body of earlier literature is fitted into this cooperative framework. Corresponding weak laws are also established, both conditionally and unconditionally, for random pooling strategies.

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Abstract  

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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Abstract  

Based on a stochastic extension of Karamata’s theory of slowly varying functions, necessary and sufficient conditions are established for weak laws of large numbers for arbitrary linear combinations of independent and identically distributed nonnegative random variables. The class of applicable distributions, herein described, extends beyond that for sample means, but even for sample means our theory offers new results concerning the characterization of explicit norming sequences. The general form of the latter characterization for linear combinations also yields a surprising new result in the theory of slow variation.

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ERDŐS, P. and RÉNYI, A., On a new law of large numbers, J. Analyse Math. 23 (1970), 103-111. MR 42 # 6907 On a new law of large numbers J. Analyse Math

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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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