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A strong law for weighted sums of i.i.d. random variables. (English) Zbl 0833.60031

Let $X$, ${X}_{1},{X}_{2},\cdots ,{X}_{n},\cdots$ be i.i.d. random variables with $𝐄X=0$ and ${S}_{n}={\sum }_{i=1}^{n}{a}_{in}{X}_{n}$, where ${\left\{{a}_{in}\right\}}_{i=1\left(1\right)n}$ is an array of constants. E.g., for the sequence $\left\{{S}_{n}\right\}$ of weighted sums a strong law is proved in the case ${sup}_{n}\sqrt[q]{\frac{1}{n}{\sum }_{i=1}^{n}{|{a}_{in}|}^{q}}<\infty ,q\in \left(1;\infty \right],$ in the following manner:

${𝐄|X|}^{p}<\infty ,\phantom{\rule{4pt}{0ex}}p:=\frac{q}{q-1},\phantom{\rule{4.pt}{0ex}}\text{implies}\phantom{\rule{4.pt}{0ex}}\frac{1}{n}{S}_{n}\to 0\phantom{\rule{4.pt}{0ex}}\text{a.s.}\phantom{\rule{4.pt}{0ex}}\text{(Theorem}\phantom{\rule{4.pt}{0ex}}\text{1.1).}$

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 B. D. Choi and S. H. Sung [Stochastic Anal. Appl. 5, 365-377 (1987; Zbl 0633.60049)].

##### MSC:
 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks
##### References:
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