Note on multidimensional empirical processes for φ-mixing random vectors

Abstract

In 1974, Sen proved weak convergence of the empirical processes (in the J 1-topology on D p [0, 1]) for a stationary φ-mixing sequence of stochastic p(≧ 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants { φ n }, i.e., φ n = O( n −1−δ) for some δ > 0.