Hitting probabilities for the Greenwood model and relations to near constancy oscillation

Abstract

We derive some properties of the Greenwood epidemic Galton–Watson branching model. Formulas for the probability $h(i,j)$ that the associated Markov chain $X$ hits state $j$ when started from state $i\ge j$ are obtained. For $j\ge1$, it follows that $h(i,j)$ slightly oscillates with varying $i$ and has infinitely many accumulation points. In particular, $h(i,j)$ does not converge as $i\to\infty$. It is shown that there exists a Markov chain $Y$ which is Siegmund dual to the chain $X$. The hitting probabilities of the dual Markov chain $Y$ are investigated.