Decomposable branching processes having a fixed extinction moment

TL;DR

This paper investigates the asymptotic probability and distribution of a decomposable critical branching process dying at a fixed moment, providing limit theorems that relate to the structure of certain random trees.

Contribution

It establishes new conditional limit theorems for the distribution of particles in decomposable critical branching processes at fixed extinction times.

Findings

Derived asymptotic probabilities for fixed extinction moments.

Proved limit theorems describing particle distributions conditioned on extinction.

Connected branching process behavior to properties of random trees.

Abstract

The asymptotic behavior, as $n\rightarrow \infty $ of the probability of the event that a decomposable critical branching process $\mathbf{Z}(m)=(Z_{1}(m),...,Z_{N}(m)),$ $m=0,1,2,...,$ with $N$ types of particles dies at moment $n$ is investigated and conditional limit theorems are proved describing the distribution of the number of particles in the process $\mathbf{Z}(\cdot)$ at moment $m<n,$ given that the extinction moment of the process is $n$. These limit theorems may be considered as the statements describing the distribution of the number of vertices in the layers of certain classes of simply generated random trees having a fixed hight.