Regenerative bootstrap for $\beta$-null recurrent Markov chains

TL;DR

This paper demonstrates the validity of two regeneration-based bootstrap methods for estimating integrals in $eta$-null recurrent Markov chains with an accessible atom, extending the CLT for randomly indexed sequences.

Contribution

It establishes the validity of regeneration-based bootstrap methods for $eta$-null recurrent Markov chains and extends the CLT for randomly indexed sequences.

Findings

Bootstrap methods are valid for $eta$-null recurrent chains.

Extension of CLT for randomly indexed sequences.

Applicable to Markov chains with accessible atoms.

Abstract

Two regeneration-based bootstrap methods, namely, the \textit{Regeneration based-bootstrap} \cite{AthreyaFuh1992, Somnat-1993} and the \textit{Regenerative Block bootstrap} \cite{Bertail2006} are shown to be valid for the problem of estimating the integral of a function with respect to the invariant measure in a $\beta$-null recurrent Markov chain with an accessible atom. An extension of the Central Limit Theorem for randomly indexed sequences is also presented.