Complete convergence for randomly indexed sums of random variables
A sequence of random variables \(\{U_n, n \geq 1\}\) is said to converge completely to a constant \(c\) if \(\sum^\infty_{n = 1} P (|U_n - c |\geq \varepsilon\} < \infty\) for all \(\varepsilon > 0\). Let \(\{X_n, n \geq 1\}\) be independent identically distributed random variables and \(S_n = X_1 + \cdots + X_n\). Let also \(\{v_n, n \geq 1\}\) be integer-valued random variables. The main result is Theorem 3. Let \(r \alpha > 1\), \(\alpha > 1/2\). Assume that \({\mathbf E} |X |^t < \infty\) for some \(t > 1/ \alpha\), and \({\mathbf E} X = 0\) if \(t \geq 1\). Assume that for some \(\beta > 0\) and \(c > 0\), \[ \sum^\infty_{n = 1} n^{r \alpha - 2} {\mathbf P} (v_n < cn^\beta) < \infty. \] If for some \(\varepsilon_0 > 0\), \(\sum^\infty_{n = 1} n^{r \alpha - 2} {\mathbf P} (\max_{k < v_n} |X_k |\geq \varepsilon_0 v^\alpha_n) < \infty\), then there exists \(\varepsilon_1 = O (\varepsilon) > 0\) such that \(\sum^\infty_{n = 1} n^{r \alpha - 2} {\mathbf P} (\max_{k < v_n} |S_{v_n} |\geq \varepsilon v^\alpha_n) < \infty\) for all \(\varepsilon \geq \varepsilon_1\). The case \(r \alpha = 1\) is considered separately. Some applications for subsequences and sequences with multidimensional indices are given.
scientific article; zbMATH DE number 897823
scientific article