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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

##### MSC:
 60F15 Strong limit theorems
Zbl 0303.60025
Full Text:
##### References:
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