Abstract: Let Yⁱ;－∞＜i＜∞ be a doubly infinite sequence of i.i.d. B-valued random variables, a_i；－∞＜i＜∞ an absolutely summable sequence of real numbers and 1≤t＜2, r＞ 1. In this paper, we prove more general version of complete convergence of \sum_k=1^n \sum_i=-\infty^\infty a_i+k Y_i / n^1 / t ; n \geq1,assuming \mathrmE Y_1=0, \mathrmE\left\|Y_1\right\|^r t<\infty, n^-1 / t \sum_k=1^n Y_k \xrightarrowp 0. In addition,when Yⁱ;-∞＜i＜∞ be a sequence of i.i.d. real random variables,we also prove more general version of complete convergence of\left\\sum_k=1^n \sum_i=-\infty^\infty a_i+k Y_i /(n \Phi(n))^1 / p ; n \geq 1\right., assuming \mathrmE Y_1=0, \mathrmE\left(\frac\left|Y_1\right|^p\Phi\left(\left|Y_1\right|^p\right)\right)^q / p<\infty, 1 \leq p<2, pS be a set of functions.