Bootstrap uniform central limit theorems for Harris recurrent Markov chains

Abstract

The main objective of this paper is to establish bootstrap uniform functional central limit theorem for Harris recurrent Markov chains over uniformly bounded classes of functions. We show that the result can be generalized also to the unbounded case. To avoid some complicated mixing conditions, we make use of the well-known regeneration properties of Markov chains. We show that in the atomic case the proof of the bootstrap uniform central limit theorem for Markov chains for functions dominated by a function in $L^{2}$ space proposed by Radulovi\'{c} (2004) can be significantly simplified. Finally, we prove bootstrap uniform central limit theorems for Fr\'{e}chet differentiable functionals in a Markovian setting.