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

MSC:
60F15 Strong limit theorems
60E15 Inequalities; stochastic orderings
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