On discrete-time semi-Markov processes

TL;DR

This paper develops a theory for discrete-time semi-Markov processes, showing they can be viewed as time-changed Markov chains and establishing their convergence to continuous-time counterparts under scaling.

Contribution

It introduces a discrete-time framework for semi-Markov processes, extending continuous-time theories and deriving governing convolution equations.

Findings

Discrete-time semi-Markov chains can be represented as time-changed Markov chains.

The paper derives convolution-type equations governing these processes.

Discrete-time processes converge weakly to continuous-time semi-Markov processes under scaling.

Abstract

In the last years, many authors studied a class of continuous time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop the discrete-time version of such a theory. We show that a class of discrete-time semi-Markov chains can be seen as time-changed Markov chains and we obtain governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.