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

In the present paper, the sufficient and necessary conditions that ∑k=1∞P(∥∑i=1nkXi∥⩾εank)<+∞ are derived for the sequence of B-valued i.i.d. random variables {Xi}, the strictly increasing subsequence of positive integers {nk} and the positive monotone sequence of real numbers {an} with an↑∞.