Extinction probability for a critical general branching process

Abstract

A general branching process begins with a single individual born at time t=0. At random ages during its random lifespan L it gives birth to offspring, N( t) being the number born in the age interval [0, t]. Each offspring behaves as a probabilistically independent copy of the initial individual. Let Z( t) be the population at time t, and let N= N(∞). Theorem: If a general branching process is critical, i. e E{ N}=1, and if σ 2=E {N(N−1)}<∞, 0<a≡ ∫ 0 ∞ tdE{N(t)} ,and as t → ∞ both t 2(1− E {N(t)})→0 and t 2 P[L>t]→0 , then tP[Z(t)>0]→ 2a σ 2 as t→∞.