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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
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