Entrance and sojourn times for Markov chains. Application to $(L,R)$-random walks

TL;DR

This paper introduces a methodology for calculating sojourn time distributions in Markov chains, using linear systems and generating functions, with applications to certain classes of bounded integer-valued random walks.

Contribution

It develops a general approach for analyzing sojourn times in Markov chains, specifically applying it to classes of bounded integer-valued random walks.

Findings

Methodology for computing sojourn time distributions using linear systems.

Application of the methodology to specific classes of random walks.

Enhanced understanding of entrance and sojourn times in Markov processes.

Abstract

In this paper, we provide a methodology for computing the probability distribution of sojourn times for a wide class of Markov chains. Our methodology consists in writing out linear systems and matrix equations for generating functions involving relations with entrance times. We apply the developed methodology to some classes of random walks with bounded integer-valued jumps.