Abstract: Let { X_j, j≥ 1} be a sequence of independant and identically distributed random variables. Let U(0,1) be a random variable distributed uniformly over (0,1) and ξ(x) be the fractional part of x. under certain conditions, ξ(max_1≤j≤nX_j) converges in distribution to U(0,1). This paper is devoted to estimation of the total variation sup_B∈B |P( ξ(max_1≤j≤nX_j)∈ B) - P(U(0,1)∈B)|。

Key words: variational distance, distribution (mod 1), extreme values

摘要： 考虑独立同分布随机变量列{X_j,j≥1}。设U(0,1)是具有(0,1)上均匀分布的随机变量，ξ(x)表示x的小数部分。适当的条件下，ξ(max_1≤j≤n X_j)依分布收敛到U(0,1)。估计全变差距离sup _B∈B| P (ξ(max_1≤j≤nX_j)∈B)-P(U(0,1)∈B) | 。

关键词: 全变差距离, 模1分布, 极值