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 0$ such that for $0 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 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 $h'' (1) 0$. 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.