The joint distribution of occupation times of skip-free Markov processes and a class of multivariate exponential distributions

TL;DR

This paper derives a simple expression for the joint Laplace transform of occupation times in skip-free Markov processes and explores conditions for these times to form a Markov chain.

Contribution

It provides a novel explicit formula for the joint distribution of occupation times and analyzes their Markovian properties.

Findings

Explicit joint Laplace transform formula for occupation times

Conditions identified for occupation times to form a Markov chain

Connection established with multivariate exponential distributions

Abstract

For a skip-free Markov process on non-negative integers with generator matrix Q, we evaluate the joint Laplace transform of the occupation times before hitting the state n (starting at 0). This Laplace transform has a very straightforward and familiar expression. We investigate the properties of this Laplace transform, especially the conditions under which the occupation times form a Markov chain.