An age dependent branching process with variable lifetime distribution

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Abstract
A branching process with variable lifetime distribution is defined by a sequence of distribution functions {G i (t)}, together with a probability generating function, h(s) = Σ k ∞= 0pks k . An ith generation particle lives a random length of time, determined by G i (t). At the end of a particle's life it produces children, the number being determined by h(s). These offspring behave like the initial particle except they are (i + 1)th generation particles and have lifetime distribution G i + 1 (t). Let Z i (t) be the number of particles alive at time t, the initial particle being born into the ith generation. Integral equations are derived for the moments of Z i (t) and it is shown that for some constants N i , γ, a, Z i (t)/(N i t γ-1 e αt ) converges in mean square to a proper random variable.