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Moment conditions for almost sure convergence of weakly correlated random variables. (English) Zbl 0785.60018
Let $\{\xi\sb k, k\in N\}$ be a real random sequence on a probability space $(\Omega,{\cal M},P)$. Define $$\widetilde\rho(k)= \sup\bigl\{\text{corr}(V;W);\ V\in L\sb 2({\cal F}\sb S),\ W\in L\sb 2({\cal F}\sb T\bigr\},$$ where ${\cal F}\sb A$ denotes the $\sigma$- field generated by $\xi\sb k$, $k\in A$, and the supremum is taken over all finite subsets $S$, $T\in N$ such that $\text{dist}(S,T)\ge k$. Further, let $\widetilde r(k)=\sup\bigl\{\text{corr}(V,W)\bigr\}$, where the supremum is taken over all finite subsets $S$, $T\subset N$ such that $\text{dist}(S,T)\ge k$ and over all linear combinations $V$ of variables $\{\xi\sb k;\ k\in S\}$ and all linear combinations $W$ of variables $\{\xi\sb k;\ k\in T\}$. The authors obtain the following two results: (i) If $\widetilde\rho(k)<1$ for some $k$, and if $E \xi\sb j=0$, $E \xi\sp 2\sb j=1$ for all $j$, $\sup\sb j E\vert \xi\sb j\vert\sp{2+\delta}<\infty$ for some $\delta$ $(>0)$ and $\sum a\sp 2\sb j<\infty$, then $\sum a\sb j\xi\sb j$ converges almost surely. (ii) If $\widetilde r(k)<1$ for some $k$, $E \xi\sb j=0$ for all $j$ and $\sum j\sp{-3/2} E \xi\sp 2\sb j<\infty$, then $n\sp{-1} \sum\sp n\sb 1 \xi\sb j\to 0$ almost surely.

60F15Strong limit theorems
60E15Inequalities in probability theory; stochastic orderings
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