On complete convergence of the sum of a random number of stable type \(p\) random elements
Let \(\{ X_ n\}\) be a sequence of independent mean-zero random elements taking values in a Banach space \(E\) of stable type \(p\), \(1 \leq p < 2\). Define \(p(E) = \text{sup} \{q : E\) is of stable type \(q\}\). Suppose that, for some random variable (r.v.) \(X\) and constant \(D\), \(\sup_{n \geq 1} P\{ \| X_ n \| > t\} \leq DP\{ X > t\}\) for every \(t > 0\), and that \(EX^ r < \infty\) for some \(p < r < p(E)\). Let \(\{\alpha_ n\}\) be an increasing positive-integer sequence and let \(\{\varphi(n)\}\) be a positive sequence such that \(\varphi (\alpha_ n) \geq a \alpha^{1/p}_ n\). Define \(S_ n = \sum^ n_{i = 1} X_ i\) and the r.v. \(N = \max \{n : \alpha_ n \leq \text{card}\{m : \varphi(m) \leq 2X\}\}\). If \(E \left( \sum^ N_{i = 1} \alpha_ i\right) < \infty\), then it is shown that \(S_{\alpha_ n}/ \varphi(\alpha_ n) \to 0\) completely. Moreover, under stronger hypotheses, it is shown that \(S_{T_ n} / \varphi(T_ n) \to 0\) completely for a class of stopping times \(\{T_ n\}\). These results extend theorems of \textit{A. Adler} [Bull., Calcutta Stat. Assoc. 37, No. 147/148, 161-169 (1988; Zbl 0689.60047)] and \textit{A. Gut} [Ann. Probab. 13, 1286-1291 (1985; Zbl 0582.60057)].
scientific article; zbMATH DE number 733291