Extinction probability for a critical general branching process

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([infinity]). Theorem: If a general branching process is critical, i. e E{N}=1, and if ,and as t --> [infinity] both t2(1-E {N(t)})-->0 and t2P[L>t]-->0, then tP[Z(t)>0]-->2a/[sigma]2 as t-->[infinity].