Occupation time of a renewal process coupled to a discrete Markov chain

TL;DR

This paper extends the Lamperti law to semi-Markov processes, providing explicit Laplace space expressions for occupation times, analyzing their long-term behavior and finite-time corrections.

Contribution

It introduces a generalized explicit formula for occupation times in semi-Markov processes, expanding classical results for Markov processes.

Findings

Derived Laplace space distribution for occupation times

Analyzed long-time limiting distribution of occupation times

Identified finite-time corrections to moments

Abstract

A semi-Markov process is one that changes states in accordance with a Markov chain but takes a random amount of time between changes. We consider the generalisation to semi-Markov processes of the classical Lamperti law for the occupation time of a two-state Markov process. We provide an explicit expression in Laplace space for the distribution of an arbitrary linear combination of the occupation times in the various states of the process. We discuss several consequences of this result. In particular, we infer the limiting distribution of this quantity rescaled by time in the long-time scaling regime, as well as the finite-time corrections to its moments.