Renewal Type Bootstrap for $\boldsymbol{U}$-process Markov Chains

S. Bouzebda , I. Soukarieh

2022, v.28, Issue 5, 673-735

ABSTRACT

The main purpose of the present paper is to establish bootstrap uniform functional central limit theorems $U$-processes for Harris recurrent Markov chains over uniformly classes of functions satisfying some entropy conditions. To simplify our approach, we will employ the well-known regenerative properties of Markov chains avoiding some complicated mixing conditions. Our result is obtained under minimal condition on the envelope function.
We next consider an extension to the $k$ Markov chain setting and prove the bootstrap consistency. The theoretical uniform central limit theorems set out below are (or will be) key tools for many further developments in Markovian data analysis.

Keywords: Bootstrap, Markov chains, regenerative processes, empirical processes, VC classes of functions, $U$-processes, Donsker classes, Central limit theorem

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