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

Let {ξ k ,kN} be a real random sequence on a probability space (Ω,,P). Define

ρ ˜(k)=supcorr (V;W) ; V L 2 ( S ) , W L 2 ( T ,

where A denotes the σ- field generated by ξ k , kA, and the supremum is taken over all finite subsets S, TN such that dist(S,T)k. Further, let r ˜(k)=supcorr ( V , W ), where the supremum is taken over all finite subsets S, TN such that dist(S,T)k and over all linear combinations V of variables {ξ k ;kS} and all linear combinations W of variables {ξ k ;kT}. The authors obtain the following two results:

(i) If ρ ˜(k)<1 for some k, and if Eξ j =0, Eξ j 2 =1 for all j, sup j E|ξ j | 2+δ < for some δ (>0) and a j 2 <, then a j ξ j converges almost surely.

(ii) If r ˜(k)<1 for some k, Eξ j =0 for all j and j -3/2 Eξ j 2 <, then n -1 1 n ξ j 0 almost surely.


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