Bootstrap uniform central limit theorems for Harris recurrent Markov chains

TL;DR

This paper establishes bootstrap uniform central limit theorems for Harris recurrent Markov chains, utilizing regeneration properties to simplify proofs and extend results to unbounded functions and differentiable functionals.

Contribution

It generalizes bootstrap uniform CLTs for Harris recurrent Markov chains to unbounded functions and differentiable functionals, simplifying proofs via regeneration techniques.

Findings

Bootstrap uniform CLT holds for bounded classes of functions.

Results extend to unbounded functions in $L^{2}$ space.

Bootstrap CLT applies to Fréchet differentiable functionals.

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.