Hitting time distribution for skip-free Markov chains: A simple proof

A well-known theorem for an irreducible skip-free Markov chain on the nonnegative integers 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.