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More on complete convergence for arrays. (English) Zbl 1087.60030
The authors consider complete convergence theorems for triangular arrays of \(\{X_{ni}, i= 1,\dots, k_n\), \(n\geq 1\}\) of rowwise independent random variables. Let \(\{a_n, n\geq 1\}\) be a sequence of positive constants such that their partial sums diverge. Under appropriate and comparatively weak conditions expressed in terms of the tail probabilities \(P(|X_{ni}|> \varepsilon)\), the expectations and variances of the truncated random variables \(X_{ni}I(|X_{ni}|\leq\delta)\), it is proved that the sums \(\sum_{n\geq 1} a_n P(|\sum^{k_n}_{i=1} X_{ni}|> \varepsilon)\) are finite for any \(\varepsilon> 0\). This paper corrects an erroneous proof of T.-C. Hu, D. Szynal and A. I. Volodin [Stat. Probab. Lett. 38, No. 1, 27–31 (1998); addendum ibid. 47, No. 2, 209–211 (2000; Zbl 0910.60017)]. It is mentioned that some of the results can be extended to triangular arrays of rowwise independent random elements taking values in a real separable Banach space.

60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
60E15 Inequalities; stochastic orderings
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
Full Text: DOI
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