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(∞). Theorem: If a general branching process is critical, i. e E{N}=1, and if $\text{σ}{}^2\text{=E {N(N?1)}<∞, 0<a≡}\text{∫}\text{0}\text{∞}\text{tdE{N(t)}}$,and as t → ∞ both $\text{t}{}^2\text{(1?}\text{E}\text{{N(t)})→0}$ and $\text{t}{}^2\text{P}\text{L>t]→0}$, then $\text{tPZ(t)>0]→}\text{2a}\text{σ}{}^2$ as t→∞.