The Annals of Mathematical Statistics

On the Supercritical One Dimensional Age Dependent Branching Processes

Krishna B. Athreya

Full-text: Open access

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.

Article information

Source
Ann. Math. Statist., Volume 40, Number 3 (1969), 743-763.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177697585

Digital Object Identifier
doi:10.1214/aoms/1177697585

Mathematical Reviews number (MathSciNet)
MR243628

Zentralblatt MATH identifier
0175.46603

JSTOR
links.jstor.org

Citation

Athreya, Krishna B. On the Supercritical One Dimensional Age Dependent Branching Processes. Ann. Math. Statist. 40 (1969), no. 3, 743--763. doi:10.1214/aoms/1177697585. https://projecteuclid.org/euclid.aoms/1177697585


Export citation