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Strong laws for weighted sums of i. i. d. random variables. (English) Zbl 1020.60016
Let $$(X_n, n\geq 1)$$ be a sequence of independent identically distributed random variables satisfying $$EX=0$$ and $$E[\exp (h |X_1|^{\gamma}) ] < \infty$$ for any $$h>0$$ $$(\gamma >0)$$. Let $$(a_{ni}$$, $$i\leq i \leq n$$, $$n\geq 1)$$ be an array of constants satisfying $${\lim \sup}_{n \to \infty} \frac{1}{n} \sum_{i=1}^n |a_{ni} |^{\alpha} < \infty$$ for some $$1 < \alpha \leq 2$$. Define $$b_n = n^{1/\alpha}(\log n)^{1/\gamma}$$ for $$0 < \gamma \leq 1$$ and $$b_n = n^{1/\alpha}(\log n) ^{1/\gamma + \delta}$$ for $$\gamma >1$$, where $$\delta =1 -1/\gamma -(\gamma -1)/ (1+ \alpha \gamma- \alpha)$$. Then $$\frac{1}{b_n} \sum_{i=1}^n a_{ni} X_i \to 0 \text{ almost surely}.$$ Also the case $$E[\exp (h |X_1|^{\gamma}) ] < \infty$$ for some $$h> 0$$ is considered.

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