Regeneration and Fixed-Width Analysis of Markov Chain Monte Carlo Algorithms

TL;DR

This thesis advances the theoretical understanding of Markov Chain Monte Carlo methods by establishing regeneration-based CLTs, relaxing variance estimation assumptions, and providing fixed-width results for adaptive algorithms.

Contribution

It introduces new regeneration-based CLTs, relaxes assumptions for variance estimation, and develops fixed-width results for MCMC estimators in general state spaces.

Findings

Regeneration approach yields necessary and sufficient conditions for CLTs.

Relaxed assumptions enable consistent variance estimation.

Fixed-width results apply to non-compact state spaces and unbounded functions.

Abstract

In the thesis we take the split chain approach to analyzing Markov chains and use it to establish fixed-width results for estimators obtained via Markov chain Monte Carlo procedures (MCMC). Theoretical results include necessary and sufficient conditions in terms of regeneration for central limit theorems for ergodic Markov chains and a regenerative proof of a CLT version for uniformly ergodic Markov chains with $E_{\pi}f^2< \infty.$ To obtain asymptotic confidence intervals for MCMC estimators, strongly consistent estimators of the asymptotic variance are essential. We relax assumptions required to obtain such estimators. Moreover, under a drift condition, nonasymptotic fixed-width results for MCMC estimators for a general state space setting (not necessarily compact) and not necessarily bounded target function $f$ are obtained. The last chapter is devoted to the idea of adaptive Monte Carlo simulation and provides convergence results and law of large numbers for adaptive procedures under path-stability condition for transition kernels.