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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.

##### MSC:
 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)
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##### References:
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