The complete convergence of subsequence for sums of independent B-valued random variables (Q2567181)

From the author's abstract: ``Sufficient and necessary conditions that \(\sum_{k=1}^{\infty} \mathbb P(\| \sum_{i=1}^{n_{k}} X_{i}\| \geq \varepsilon a_{n_{k}}) < \infty\) are derived for the sequence of B-valued i.i.d. random variables \(\{ X_{i} \},\) the strictly increasing subsequence of positive integers \(\{ n_{k} \}\) and the positive monotone sequence of real numbers \(\{ a_{n} \}\) with \(a_{n} \to \infty.\)'' In fact, equivalence is proved under the additional assumption \(\limsup_{k \to \infty} n_{k+1}/\Big(\sum_{i=1}^{k} n_{i}\Big) < \infty\) and some other technical assumptions relating the growth of the subsequence \(a_{n_k}\) to the growth of \(n_{k}.\) This extends some previously known results in the real-valued case [\textit{S. Asmussen} and \textit{T. G. Kurtz}, Ann. Probab. 8, 176--182 (1980; Zbl 0426.60026); \textit{K.B. Athreya} and \textit{N. Kaplan}, ibid. 4, 38--50 (1976; Zbl 0356.60048) and in: Branching processes. Adv. Probab. relat. Top., Vol. 5, 27--60 (1978; Zbl 0404.60088); \textit{O. Nerman}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 57, 365--395 (1981; Zbl 0451.60078); \textit{A. Gut}, Ann. Probab. 13, 1286--1291 (1985; Zbl 0582.60057); \textit{Q. Wang} and \textit{C. Su}, J. Math., Wuhan Univ. 11, No.~2, 161--171 (1991; Zbl 0749.60032); \textit{D. Deng}, Acta Math. Appl. Sin. 16, No.~3, 308--316 (1993; Zbl 0781.60026)]. The main approach is due to the Talagrand's isoperimetric inequality and entropy estimate.