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

Two regeneration-based bootstrap methods, namely, the $\mathit{\text{Regeneration}}\mathit{\text{based-bootstrap}}$ \cite{AthreyaFuh1992, Somnat-1993} and the $\mathit{\text{Regenerative}}\mathit{\text{Block}}\mathit{\text{bootstrap}}$ \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.