Asymptotic Properties of an Age Dependent Branching Process

Abstract

Let $Z(t)$ denote the number of cells at time $t$ which are progeny of a single cell born at $t = 0, G(t)$ with $G(0) = 0$ be the lifetime distribution function of each cell, and $h(s) = \sum^\infty_{r = 0} s^r$, where $p_r$ are constants, $p_r \geqq 0, \sum^\infty_{r = 0} p_r = 1$ be the generating function of the number of cell progeny which replace each cell on completion of its life. Cells develop and proliferate independently of each other. For general $G(t)$ this process is called an age dependent branching process and for $G(t)$ an exponential distribution, a Markov branching process [3]. When the mean number of progeny per cell, $h^{(1)} (1) = 1$, and $h^{(2)} (1) > 0, h^{(3)} (1) < \infty$, and $G(t)$ is an exponential distribution with parameter $\lambda$, Sevast'yanov [5] showed by study of a differential equation satisfied by $F(s, t) = \sum^\infty_{j = 0} P\lbrack Z(t) = j\rbrack s^j$, that $\lim_{t\rightarrow\infty} tP\lbrack Z(t) > 0\rbrack = 2\lbrack\lambda h^{(2)} (1)\rbrack^{-1}$ and that for $u \geqq 0$, \begin{equation*}\tag{1}\lim_{t\rightarrow\infty} P\lbrack 2(\lambda h^{(2)} (1)t)^{-1}Z(t) > u \mid Z(t) > 0\rbrack = \exp (-u).\end{equation*} Analogous limit theorems for the discrete time case were obtained by Kolmogorov and by Yaglom. See [3], pp. 21-22, 108-109. It is the purpose of this paper to extend the results of Sevast'yanov to the case of general $G(t)$. In Section 1, Theorem 1 gives the form of the asymptotic moments of such an age dependent branching process by study of an integral equation satisfied by $D(s, t) = 1 - E \lbrack\exp (-sZ(t))\rbrack$. Chover and Ney [1] have shown that for mild conditions on $G(t)$ and $h(s)$, that $\lim_{t\rightarrow\infty} tP\lbrack Z(t) > 0\rbrack = b$, where $b$ is a strictly positive constant to be defined. In Section 2, this result, together with Theorem 1 yields a conditional limit theorem which generalizes (1). Section 3 contains remarks on an analogous general discrete time result of Mullikin [4].