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A strong law for weighted sums of i.i.d. random variables. (English) Zbl 0833.60031
Let $X$, $X_1, X_2, \dots, X_n, \dots$ be i.i.d. random variables with ${\bold E}X = 0$ and $S_n = \sum^n_{i = 1} a_{in} X_n$, where $\{a_{in}\}_{i = 1(1)n}$ is an array of constants. E.g., for the sequence $\{S_n\}$ of weighted sums a strong law is proved in the case $\sup_{n} \root q \of {{1\over n} \sum^n_{i = 1} |a_{in}|^q} < \infty, q\in (1;\infty],$ in the following manner: $${\bold E} |X|^p < \infty,\ p := {q\over q -1}, \text{ implies } {1\over n} S_n \to 0 \text{ a.s. (Theorem 1.1).}$$ Also the case $q = 1$ is investigated. Extensions to more general normalizing sequences are given and necessary and sufficient conditions are discussed. Several well-known results are contained, e.g. in the case $q = \infty$, i.e. $\text{sup} |a_{in}|< \infty$ and $p = 1$, the result by {\it B. D. Choi} and {\it S. H. Sung} [Stochastic Anal. Appl. 5, 365-377 (1987; Zbl 0633.60049)].

60F15Strong limit theorems
60G50Sums of independent random variables; random walks
Full Text: DOI
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