Sums of Independent Random Variables on Partially Ordered Sets

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.