Hitting Time Distribution for Skip-Free Markov Chains: A Simple Proof

TL;DR

This paper provides a straightforward proof for the distribution of hitting times in skip-free Markov chains, showing they can be expressed as sums of independent geometric or exponential variables.

Contribution

It introduces a simple, direct proof for the known distribution theorem of hitting times in both discrete and continuous skip-free Markov chains using generating functions.

Findings

Hitting times are distributed as sums of independent geometric/exponential variables.

The proof uses direct calculation of generating functions or Laplace transforms.

Applicable to both discrete and continuous time skip-free Markov chains.

Abstract

A well-known theorem for an irreducible skip-free chain with absorbing state $d$, under some conditions, is that the hitting (absorbing) time of state $d$ starting from state 0 is distributed as the sum of $d$ independent geometric (or exponential) random variables. The purpose of this paper is to present a direct and simple proof of the theorem in the cases of both discrete and continuous time skip-free Markov chains. Our proof is to calculate directly the generation functions (or Laplace transforms) of hitting times in terms of the iteration method.