Abstract
It is well known that a supercritical single-type Bienayme-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants. In this paper we analyze such a decomposition for the linear-fractional Bienayme-Galton-Watson processes with countably many types. We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles.
Highlights
The Bienaymé-Galton-Watson (BGW-) process is a basic model for the stochastic dynamics of the size of a population formed by independently reproducing particles
It is well known that a supercritical single-type Bienaymé-Galton-Watson process can be viewed as a decomposable branching process formed by two subtypes of particles: those having infinite line of descent and those who have finite number of descendants
We find explicit expressions for the main characteristics of the reproduction laws for so-called skeleton and doomed particles
Summary
The Bienaymé-Galton-Watson (BGW-) process is a basic model for the stochastic dynamics of the size of a population formed by independently reproducing particles. It has a long history [1] with its origin dating back to. This paper is devoted to the BGW-processes with countably many types. If we disregard the doomed particles, the skeleton particles form a BGW-process with a transformed reproduction law excluding extinction f s f s 1 q q q (3). In the special case when the reproduction generating function (1) is linear-fractional many characteristics of the BGW-process can be computed in an explicit form [6]. This paper presents a case where the properties of the skeleton and doomed particles are very transparent
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