Submartingale Condition for Weak Convergence for Semi-Markov Processes

TL;DR

This paper adapts a submartingale condition for weak convergence to semi-Markov processes, highlighting the need for an additional condition when using an embedded Markov chain and jump-time filtration.

Contribution

It introduces a modified submartingale condition specific to semi-Markov processes, showing the classical condition's limitations and proposing necessary adjustments.

Findings

Classical submartingale condition is not directly applicable to semi-Markov processes.

An additional condition is required for weak convergence in the semi-Markov setting.

The paper clarifies the role of embedded Markov chains and jump-time filtrations in the convergence analysis.

Abstract

In this paper, we consider a modified version of a well-known submartingale condition fortheweak convergence of probabilitymeasures, adapted to the semi-Markov case. In this setting, it is convenient to work with an embedded Markov chain and the filtration generated by jump times. We demonstrate that a straightforward restatement of the classical result is not valid, and that an additional condition is required.