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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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References:
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