Weak convergence for stationary bootstrap empirical processes of associated sequences

Abstract

In this work the stationary bootstrap of Politis and Romano \cite{PR1994a} is applied to the empirical distribution function of stationary and associated random variables. A weak convergence theorem for the stationary bootstrap empirical processes of associated sequences is established with its limiting to a Gaussian process almost surely, conditionally on the stationary observations. The weak convergence result is proved by means of a random central limit theorem on geometrically distributed random block size of the stationary bootstrap procedure. As its statistical applications, stationary bootstrap quantiles and stationary bootstrap mean residual life process are discussed. Our results extend the existing ones of Peligrad \cite{P1998} who dealt with the weak convergence of non-random blockwise empirical processes of associated sequences as well as of Shao and Yu \cite{SY1996} who obtained the weak convergence of the mean residual life process in reliability theory as an application of the association.