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Moment conditions for almost sure convergence of weakly correlated random variables. (English) Zbl 0785.60018

Let $\left\{{\xi }_{k},k\in N\right\}$ be a real random sequence on a probability space $\left({\Omega },ℳ,P\right)$. Define

$\stackrel{˜}{\rho }\left(k\right)=sup\left\{\text{corr}\left(V;W\right);\phantom{\rule{4pt}{0ex}}V\in {L}_{2}\left({ℱ}_{S}\right),\phantom{\rule{4pt}{0ex}}W\in {L}_{2}\left({ℱ}_{T}\right\},$

where ${ℱ}_{A}$ 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 $\text{dist}\left(S,T\right)\ge k$. Further, let $\stackrel{˜}{r}\left(k\right)=sup\left\{\text{corr}\left(V,W\right)\right\}$, where the supremum is taken over all finite subsets $S$, $T\subset N$ such that $\text{dist}\left(S,T\right)\ge k$ and over all linear combinations $V$ of variables $\left\{{\xi }_{k};\phantom{\rule{4pt}{0ex}}k\in S\right\}$ and all linear combinations $W$ of variables $\left\{{\xi }_{k};\phantom{\rule{4pt}{0ex}}k\in T\right\}$. The authors obtain the following two results:

(i) If $\stackrel{˜}{\rho }\left(k\right)<1$ for some $k$, and if $E{\xi }_{j}=0$, $E{\xi }_{j}^{2}=1$ for all $j$, ${sup}_{j}E{|{\xi }_{j}|}^{2+\delta }<\infty$ for some $\delta$ $\left(>0\right)$ and $\sum {a}_{j}^{2}<\infty$, then $\sum {a}_{j}{\xi }_{j}$ converges almost surely.

(ii) If $\stackrel{˜}{r}\left(k\right)<1$ for some $k$, $E{\xi }_{j}=0$ for all $j$ and $\sum {j}^{-3/2}E{\xi }_{j}^{2}<\infty$, then ${n}^{-1}{\sum }_{1}^{n}{\xi }_{j}\to 0$ almost surely.

##### MSC:
 60F15 Strong limit theorems 60E15 Inequalities in probability theory; stochastic orderings