A general result of the almost sure central limit theorem of uniform empirical process

Abstract

Let {X_n; n≥1} be a sequence of independent and identically distributed U[0, 1]-distributed random variables. Define the uniform empirical process $F_n(t)=n^{-1/2}\sum _{i=1}^n(I_{\{X_i\le t\}}-t)$, 0≤t≤1, and |F_n|=sup _0≤t≤1|F_n(t)|. This paper is concerned with a general case of the almost sure central limit theorem of |F_n| and sup _0≤t≤1F_n(t).

Keywords:

• almost sure central limit theorem
• uniform empirical process
• Brownian bridge

Acknowledgement

Thanks are due to professor Zhengyan Lin of Zhejiang University and Zuoxiang Peng of Southwest University in China for their help. This work is supported by the National Natural Science Foundation of China (grant 11326174), Natural Science Foundation of Jiangsu Province (grant BK20130412), Natural Science Research Project of Ordinary Universities in Jiangsu Province (grant 12KJB110003).