Regeneration-enriched Markov processes with application to Monte Carlo

TL;DR

This paper introduces a class of Markov processes with regeneration mechanisms that enhance flexibility and simplify analysis for Monte Carlo sampling, providing conditions for convergence, CLT, and exact sampling.

Contribution

It develops a framework for regeneration-enriched Markov processes with conditions ensuring invariant measures, CLT, and exact sampling, improving Monte Carlo methods.

Findings

Processes can have a target invariant distribution with regeneration.

Conditions for a central limit theorem are established.

Exact sampling is possible via coupling from the past.

Abstract

We study a class of Markov processes that combine local dynamics, arising from a fixed Markov process, with regenerations arising at a state-dependent rate. We give conditions under which such processes possess a given target distribution as their invariant measures, thus making them amenable for use within Monte Carlo methodologies. Since the regeneration mechanism can compensate the choice of local dynamics, while retaining the same invariant distribution, great flexibility can be achieved in selecting local dynamics, and the mathematical analysis is simplified. We give straightforward conditions for the process to possess a central limit theorem, and additional conditions for uniform ergodicity and for a coupling from the past construction to hold, enabling exact sampling from the invariant distribution. We further consider and analyse a natural approximation of the process which may arise in the practical simulation of some classes of continuous-time dynamics.