Revisiting conditional limit theorems for the mortal simple branching process

The various conditional limit theorems for the simple branching process an considered within a unified setting. In the subcritical case conditioning e vents have the form {H is an element of n + L}, where H is the time to exti nction and L is a subset of the natural numbers. The resulting limit theore ms contain all known forms, and collectively they are equivalent to the cla ssical Yaglom form. In the critical case discrete limits exist provided L i s a finite set. The principal results are extended to absorbing Markov chai ns. The Yaglom and Harris theorems for the critical case are generalized by considering the joint behaviour of generation sizes and total progeny cond itioned by one-parameter families of events of the form {n < H less than or equal to an) and (H > an}, where 1 less than or equal to a less than or eq ual to infinity. A simple representation of the marginal limit laws of the population sizes relates the Yaglom and Harris limits. Analagous structure is elucidated for the marginal limit laws of the total progeny.