On the Supercritical One Dimensional Age Dependent Branching Processes

Abstract

Let $\{Z(t); t \geqq 0\}$ be a one dimensional age dependent branching process with offspring probability generating function (pgf) $h(s) \equiv \sum^\infty_{j=0} p_js^j$ and lifetime distribution function $G(t)$ (see Section 2 for definitions). If $m(t) \equiv EZ(t)$ is the mean function let $Y(t) = Z(t)/m(t)$. Our objective in this paper is to study the limiting behavior of the process $\{Y(t); t \geqq 0\}$. The main result is THEOREM 0. Assume $Z(0) \equiv 1, m = h'(1) > 1, G(0+) = 0$. (Here $\rightarrow_p$ and $\rightarrow_d$ mean convergence in probability and distribution respectively). Then: \begin{equation*}\tag{1}\sum^\infty_{j=2} j \log jp_j = \infty\quad\text{implies} Z(t)/EZ(t) \rightarrow_p 0\end{equation*} and \begin{equation*}\tag{2}\sum^\infty_{j=2} j \log jp_j < \infty\quad\text{implies} Z(t)/EZ(t) \rightarrow_d W\end{equation*} where $W$ is an nonnegative random variable such that (a) $EW = 1$, (b) $\varphi(u) = E(e^{-uW})$ for $u \geqq 0$ satisfies \begin{equation*}\tag{3}\varphi(u) = \int^\infty_0 h(\varphi(ue^{-\alpha y})) dG(y)\end{equation*} where $\alpha$ is the unique root of the equation $m \int^\infty_0 e^{-\alpha y} dG(y) = 1$ (c) $P(W = 0) = q$ the extinction probability (d) $W$ has an absolutely continuous distribution on the positive real axis and the density function is continuous. That is, there exists a nonnegative continuous function $g(x)$ defined for $x > 0$ such that for $0 < x_1 < x_2 < \infty$ \begin{equation*}\tag{4}P(x_1 < W < x_2) = \int^{x_2}_{x_1} g(x) dx.\end{equation*} Kesten and Stigum [4] proved the above result for the case when $G(x)$ is the step function \begin{equation*}\tag{5}\begin{align*}G(x) = 0 \text{if} x\leqq 1 \\ = 1\quad x > 1.\end{align*}\end{equation*} This is the Galton-Watson process in discrete time. They considered the multi-dimensional case. Athreya and Karlin [1] considered the case (here $0 < \lambda < \infty)$ \begin{equation*}\tag{6}\begin{align*}G(x) = 1 - e^{-\lambda x} \text{for} x > 0 \\ = 0\quad x \leqq 0.\end{align*}\end{equation*} This is the continuous time Markov branching process. Their approach was via split times. Levinson [6] established the law convergence of $Z(t)/EZ(t)$ under conditions slightly stronger than ours. Harris [3] claimed mean square convergence of $Z(t)/EZ(t)$ when $h'' (1) < \infty$ and the absolute continuity of $W$ when in addition to $h'' (1) < \infty, 1 - G(t) = O(e^{-cr})$ for some $c > 0$. Our result is the sharpest known in this direction in as much as (i) we establish the convergence of $Z(t)/EZ(t)$ without any conditions, (ii) we give a necessary and sufficient condition for the nondegeneracy of the limit random variable $W$ and (iii) when $W$ is nondegenerate we establish the absolute continuity without any extra assumptions. The methods employed in this paper are all extremely simple. Among them are a simplified and sharpened form of Levinson's [6] arguments and a simplification of Stigum's [7] idea to prove absolute continuity of $W$. One of the important ideas used here is the exploitation of the underlying Galton-Watson process constituted by the size $\{\zeta_n\}$ of the different generations. The key to the understanding of the moment condition $\sum_j j \log jp_j < \infty$ is the simple Lemma 1. Here is an outline of the rest of the paper. In Section 2 we describe the setting and introduce the necessary terminology and notation. The functional equation (3) is studied in detail in Section 3 where it is shown that a necessary and sufficient condition for (3) to have a nontrivial solution is the finiteness of $\sum j \log jp_j$. The next section explores the connection between the process $\{Z(t); t \geqq 0\}$ and the underlying Galton-Watson process $\{\zeta_n; n = 0, 1, 2, \cdots\}$ and shows that if $\sum j \log jp_j = \infty$ then $Z(t)/EZ(t) \rightarrow_p 0$. Assuming $\sum j \log jp_j < \infty$ the convergence in distribution of $Z(t)/EZ(t)$ to a nondegenerate random variable $W$ is shown in Section 5 while Section 6 takes up the proof of absolute continuity. The last section lists some open problems.