Acta Scientiarum Naturalium Universitatis Pekinensis

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The Variational Distance for Asymptotic Distribution of Extreme Values (Modulo One)

QI Yongcheng   

  1. Department of Probability & Statistics, Peking University, Beijing, 100871
  • Received:1995-12-20 Online:1997-01-20 Published:1997-01-20



  1. 北京大学概率统计系,北京,100871

Abstract: Let { Xj, 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(max1≤j≤nXj) converges in distribution to U(0,1). This paper is devoted to estimation of the total variation supB∈B |P( ξ(max1≤j≤nXj)∈ 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{Xj,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(max1≤jn Xj)依分布收敛到U(0,1)。估计全变差距离sup B∈B| P (ξ(max1≤jnXj)∈B)-P(U(0,1)∈B) | 。

关键词: 全变差距离, 模1分布, 极值

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