Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

• Published in: Science China. Mathematics 2002-02, Vol.45 (2), p.223-232
• Main Author: Zhang, Dixin
• Format: Article
• Language: English
• Subjects: Convergence
• ISSN: 1869-1862; 1674-7283; 1869-1862
• DOI: 10.1360/02ys9023

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k−r), 0 < r ⩽ 1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(1+r)/log n)−1/2) or O((nr/(1+r)/log n)−3/4) and that the Central Limit Theorem does not always hold for the empirical processes.