Supercritical general branching processes conditioned on extinction are subcritical

TL;DR

This paper extends the understanding of branching processes by proving that general, multi-type processes conditioned on extinction behave subcritically, generalizing known results from simple models.

Contribution

It demonstrates that the subcritical behavior under extinction conditioning applies to complex, multi-type branching processes with age-dependent reproduction and lifespan.

Findings

Conditioned multi-type processes are subcritical

Results generalize classical single-type models

Applicable to age-dependent and complex reproductive scenarios

Abstract

It is well known that a simple, supercritical Bienaym\'e-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction.