The Total Number of Particles in a Reduced Bellman-Harris Branching Process

Abstract

Let $z(t)$ be the number of particles at time t in a Bellman-Harris branching process with generating function $f(s)$ of the number of direct descendants and distribution $G(t)$ of particle lifelength satisfying the conditions \[f'(1) = 1,\qquad f(s) = s + (1 - s)^{1 + \alpha } L(1 - s),\] where $\alpha \in ( {0,1} ]$, the function $L(x)$ varies slowly as $x \to 0 + $, and \[\mathop {\lim }\limits_{n \to \infty } \frac{{n\left( {1 - G\left( n \right)} \right)}} {{1 - f_n \left( 0 \right)}} = 0,\] where $f_n ( s )$ is the nth iteration of $f(s)$. Denote by $\{ z(\tau ,t), 0 \leq \tau \leq t\}$ the corresponding reduced Bellman-Harris branching process, where $z(\tau ,t)$ is the number of particles in the initial process at time $\tau $ whose descendants or they themselves are alive at time t. Let $\nu (t)$ be the number of dead particles of the reduced process to time t. The paper finds the limiting distribution of $\nu(t)$ under the conditions $z(t) > 0$ and $t \to \infty $.