Branching Processes with Final Types of Particles and Random Trees

Abstract

This paper considers a Bellman–Harris branching process whose probability generating function $f(s)$ of the number of direct descendants of particles satisfies the relation $f(s) = s + (1 - s)^{1 + \alpha } L(1 - s),0 < \alpha \leqq 1$. Let $\tau $ be the moment of extinction of the process and let $\nu_\Delta $ be the total number of particles the number of direct descendants of each of which belongs to the set $\Delta ,\Delta \subset \{ 0,1, \ldots ,n, \ldots \} $. The paper gives conditions under which, for any $x \in ( - \infty , + \infty )$ and some scaling constants $b(N)$, a nondegenerate limit, $\lim _{N \to \infty } {\bf P}\{ \tau b(N) \leqq x|\nu_\Delta = N\} $, exists.