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October, 1974 Sums of Independent Random Variables on Partially Ordered Sets
R. T. Smythe
Ann. Probab. 2(5): 906-917 (October, 1974). DOI: 10.1214/aop/1176996556


Let $(\mathscr{A}, \leqq)$ be a partially ordered set, $\{X_\alpha\}_{\alpha\in\mathscr{A}}$ a collection of i.i.d. random variables with mean zero, indexed by $\mathscr{A}$. Let $S_\beta = \sum_{\alpha\leqq\beta} X_\alpha, |\beta| = \operatorname{card} \{\alpha\in\mathscr{A}: \alpha \leqq \beta\}$. We study the a.s. convergence to zero of $Z_\beta = S_\beta/|\beta|$, when $|\beta| \mapsto \infty$. We first derive a Hajek-Renyi inequality for $K^r = \{(k_1, k_2, \cdots, k_r): k_i$ a positive integer$\}$. This is used to derive a sufficient condition for the convergence of $Z_\beta$ for a class of partially ordered sets including $K^r$. For many of these sets (and certain other sets as well) this condition is shown to be necessary. Finally a weaker sufficient condition is derived for a much larger class of sets, giving a theorem analogous to one of Hsu and Robbins for the linearly ordered case.


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R. T. Smythe. "Sums of Independent Random Variables on Partially Ordered Sets." Ann. Probab. 2 (5) 906 - 917, October, 1974.


Published: October, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0292.60081
MathSciNet: MR358973
Digital Object Identifier: 10.1214/aop/1176996556

Primary: 60G50
Secondary: 60F15 , 60G45

Keywords: Independent random variables , martingale , regularly varying function , Strong law of large numbers

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 5 • October, 1974
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