A strong law for weighted sums of i.i.d. random variables
Let \(X\), \(X_1, X_2, \dots, X_n, \dots\) be i.i.d. random variables with \({\mathbf E}X = 0\) 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: \[ {\mathbf E} |X|^p < \infty,\;p := {q\over q -1}, \text{ implies } {1\over n} S_n \to 0 \text{ a.s. (Theorem 1.1).} \] 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 \textit{B. D. Choi} and \textit{S. H. Sung} [Stochastic Anal. Appl. 5, 365-377 (1987; Zbl 0633.60049)].
scientific article; zbMATH DE number 806432