What question did this study set out to answer?

This research aims to adapt a submartingale condition for weak convergence to semi-Markov processes.

February 26, 2026Modern Stochastics Theory and Applications0 citationsOpen Access

Submartingale condition for weak convergence for semi-Markov processes

VGVitaliy GolomoziyTaras Shevchenko National University of Kyiv

Key Points

This research aims to adapt a submartingale condition for weak convergence to semi-Markov processes.
Adaptation of classical results to the semi-Markov framework
Utilization of embedded Markov chains
Analysis of filtration generated by jump times
A classical submartingale condition for weak convergence does not hold in this setting
An additional condition is necessary for valid convergence in semi-Markov processes

Abstract

In this paper, we consider a modified version of a well-known submartingale condition for the weak convergence of probability measures, 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.

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