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

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.

💡 Research Summary

The dissertation develops a unified theoretical framework for analyzing Markov chain Monte Carlo (MCMC) algorithms by exploiting the split‑chain (regeneration) construction. The first part establishes necessary and sufficient regeneration conditions for central limit theorems (CLTs) of ergodic Markov chains. In particular, it shows that for uniformly ergodic chains with only a finite second moment under the stationary distribution ($E_{\pi}f^{2}<\infty$), a regenerative proof of the CLT can be given without invoking stronger mixing assumptions. This result clarifies the exact relationship between the existence of regeneration times and asymptotic normality.

The second part tackles the construction of fixed‑width confidence intervals for MCMC estimators. A fixed‑width interval requires a pre‑specified error tolerance $\epsilon$ and confidence level $1-\alpha$, together with a sample size $n$ that guarantees the interval’s coverage. The key technical obstacle is the need for a strongly consistent estimator of the asymptotic variance $\sigma^{2}$. The author proposes a variance estimator that directly uses the lengths and frequencies of regeneration blocks, thereby avoiding the bias inherent in batch‑means or spectral methods. Under a $V$‑geometric drift condition, the analysis does not require a compact state space nor a bounded test function $f$. Provided the drift function $V$ is $\pi$‑integrable and the expected regeneration length is finite, a non‑asymptotic bound of the form
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