Some probabilistic inequalities for sums of random elements (Q912467)

Let \(X_ k\), \(k=1,2,...\), be independent random elements in a separable Hilbert space H. Assume that \(E X_ k=0\) and \(\| X_ k\| \leq C E \| X_ k\|\) a.s., \(C>0\), \(k=1,2,... \). The paper is devoted to the study of probability inequalities of the Bernstein-Prokhorov-type [cf. \textit{Yu. V. Prokhorov}, Theory Probab. Appl. 4, 201-203 (1960); translation from Teor. Veroyatn. Primen. 4, 211-214 (1959; Zbl 0093.151)] for the infinite sum \(X_ 1+X_ 2+...\), if the additional condition \[ \sum^{\infty}_{k=1}E \| X_ k\|^{\alpha}<\infty \] for some \(\alpha,\) \(1<\alpha \leq 2,\) is fulfilled. Also, a generalization to unbounded random elements is obtained: \[ P(\| \sum^{\infty}_{k=1}a_ kX_ k\| >x)\leq \exp (- C_{\alpha}x^{\alpha /(\alpha -1)}),\quad C_{\alpha}>0. \] (Here we omit the exact assumptions).