Wild regenerative block bootstrap for Harris recurrent Markov chains

We consider Gaussian and bootstrap approximations for the supremum of additive functionals of aperiodic Harris recurrent Markov chains. The supremum is taken over a function class that may depend on the sample size, which allows for non-Donsker settings; that is, the empirical process need not have a weak limit in the space of bounded functions. We first establish a non-asymptotic Gaussian approximation error, which holds at rates comparable to those for sums of high-dimensional independent or one-dependent vectors. Key to our derivation is the Nummelin splitting technique, which enables us to decompose the chain into either independent or one-dependent random blocks. Additionally, building upon the Nummelin splitting, we propose a Gaussian multiplier bootstrap for practical inference and establish its finite-sample guarantees in the strongly aperiodic case. Finally, we apply our bootstrap to construct a uniform confidence band for an invariant density within a certain class of diffusion processes.