Revisiting conditional limit theorems for the mortal simple branching process

Abstract

The various conditional limit theorems for the simple branching process are considered within a unified setting. In the subcritical case conditioning events have the form $\{calH\in n+calS\}$, where $calH$ is the time to extinction, and $calS$ is a subset of the natural numbers. The resulting limit theorems contain all known forms, and collectively they are equivalent to the classical Yaglom form. In the critical case discrete limits exist provided $calS$ is a finite set. The principal results are extended to absorbing Markov chains. The Yaglom and Harris theorems for the critical case are generalized by considering the joint behaviour of generation sizes and total progeny conditioned by one-parameter families of events of the form $\{n<calH\le \alpha n\}$ and $\left\{calH>\alpha n\right\}$, where $1\le \alpha \le \mathrm{\infty }$. A simple representation of the marginal limit laws of the population sizes relates the Yaglom and Harris limits. Analogous structure is elucidated for the marginal limit laws of the total progeny.