Concatenation of Markov processes

TL;DR

This paper explores how concatenating Markov processes can be used for Monte Carlo integration, establishing conditions for their Markov property and invariance, and identifying their generators to support new Monte Carlo methods.

Contribution

It provides the first theoretical foundation for using concatenated Markov processes in Monte Carlo algorithms by identifying their generators and conditions for invariance.

Findings

Established mild conditions for concatenated processes to be Markovian.

Proved invariance of concatenated processes with respect to target distributions.

Identified the generator of concatenated Markov processes.

Abstract

We investigate the concatenation of Markov processes. Our primary concern is to utilize processes constructed in this manner for Monte Carlo integration. To enable this using conventional methods, it is essential to demonstrate the Markov property and invariance with respect to a given target distribution. We provide mild sufficient conditions for this. Our main result is the identification of the generator of the concatenation of Markov processes. This result provides the theoretical foundation for Monte Carlo methods based on this construction.