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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:
  Bai, Z.D.; Cheng, P.E., Marcinkiewicz strong laws for linear statistics, Statist. probab. lett., 46, 105-112, (2000) · Zbl 0960.60026  Choi, B.D.; Sung, S.H., Almost sure convergence theorems of weighted sums of random variables, Stochastic anal. appl., 5, 365-377, (1987) · Zbl 0633.60049  Cuzick, J., A strong law for weighted sums of i.i.d. random variables, J. theoret. probab., 8, 625-641, (1995) · Zbl 0833.60031  Hsu, P.L.; Robbins, H., Complete convergence and the law of large numbers, Proc. nat. acad. sci. USA, 33, 25-31, (1947) · Zbl 0030.20101  Rosalsky, A.; Sreehari, M., On the limiting behavior of randomly weighted partial sums, Statist. probab. lett., 40, 403-410, (1998) · Zbl 0937.60014  Stout, W.F., Almost sure convergence., (1974), Academic Press New York · Zbl 0165.52702  Wu, W.B., On the strong convergence of a weighted sum, Statist. probab. lett., 44, 19-22, (1999) · Zbl 0951.60027
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