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

Abstract

Assume that { X n } is a strictly stationary β -mixingrandom sequence with the β -mixing coefficient β k =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 L p ,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 (( n r /(1+ r ) /log n ) -1/2 ) or O ( n r /(1+ r ) /log n ) -3/4 ) and that the Central Limit Theorem does not always hold for the empirical processes.