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Chen, P.; Sung, S. H.

Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables. (English) Zbl 1374.60040

Acta Math. Hung. 148, No. 1, 83-95 (2016).
Let \(\{X_{n},n\geq 1\}\) be a sequence of identically distributed negatively orthant dependent random variables. Let \(\{a_{ni},1\leq i\leq n,n\geq 1\}\) be an array of constants such that \(\sum_{i=1}^{n}\left| a_{ni}\right| ^{\alpha}=O(n),\) for some \(0<\alpha <2.\) Let us put \( b_{n}=n^{1/\alpha}(\log n)^{1/\gamma},\) for some \(\gamma >\alpha .\) The main result of this paper presents necessary and sufficient conditions under which, for every \(\varepsilon >0,\) \(\sum_{n=1}^{\infty}n^{-1}P(\max_{1\leq m\leq n}\) \(\left| \sum_{i=1}^{m}a_{ni}X_{i}\right| >\varepsilon b_{n})<\infty .\) As a consequence of this result, a strong law of large numbers for weighted sums of negatively orthant dependent random variables is presented, too.
Reviewer: Zdzisław Rychlik (Lublin)

Cited in 15 Documents

MSC:

60F15 Strong limit theorems

Keywords:

complete convergence; weighted sum; strong law of large numbers; negatively orthant dependent random variable; Rosenthal inequality
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References:

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