Abstract: Let {X_j, j<=1} be a sequence of i.i.d. random variables, and ｛Ｓ_n｝be its partial sum sequence, and ξ（Ｓ_n）denote the fractional part of Ｓ_n. This paper estimates the convergence rate to asymptotic uniformity for ξ(Ｓ_n) i.e. supB∈B［０，１）｜P(ξ(S_n)∈B)-P(U∈B)｜.

Key words: distribution modulo1, asymptotic uniformity, convergence rate, cumulant of order k

摘要： 设｛Ｘj｝是独立同分布随机变量序列，｛Ｓn｝是其部分和序列，ξ（Ｓn）表示Ｓn的小数部分，本文讨论了ξ（Ｓn）渐近Ｕ［０，１）均匀分布的收敛速度，即估计supB∈B［０，１)｜P(ξ(Ｓn)∈B)-P(U∈Ｂ)｜。

关键词: 模1(mod1)分布, 渐近均匀性, 收敛速度, k阶累积量