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Almost sure convergence theorems of weighted sums of random variables. (English) Zbl 0633.60049
Let \(\{X,X_ n\), \(n\geq 1\}\) be random variables and \(\{a_{ni}\), \(1\leq i\leq n\), \(n>1\}^ a \)triangular array of constants. Various conditions on \(\{a_{ni}\}\) and \(\{X_ n\}\) are given, under which \(\sum^{n}_{i=1}a_{ni}X_ i\to 0\) almost surely, as \(n\to 0.\)
In particular, if the \(X_ n\), \(n\geq 1\), are identically distributed, it is sufficient that \(E| X|^ r\) and \(\sum^{n}_{i=1}a^ 2_{ni}=O(1/n^{2/r})\) for some \(r\in (0,2).\)
This improves the result by Y. S. Chow and T. L. Lai, Ann. Probab. 1, 810-824 (1973; Zbl 0303.60025). In the i.i.d. case, E X\(=0\) and \(\max_{1\leq i\leq n}| a_{ni}| =O(1/n)\) are sufficient.
Reviewer: H.Hering

60F15 Strong limit theorems
Zbl 0303.60025
Full Text: DOI
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