Extinction of decomposable branching processes

TL;DR

This paper investigates the asymptotic distribution of particle counts in a decomposable critical branching process conditioned on extinction at a large time, revealing detailed probabilistic behavior near extinction.

Contribution

It provides new asymptotic results for the distribution of particles in decomposable critical branching processes conditioned on extinction time.

Findings

Asymptotic distribution of particle counts near extinction time

Behavior of decomposable critical branching processes as extinction approaches

Conditional distribution results for large extinction times

Abstract

The asymptotic behavior, as $n\rightarrow \infty $ of the conditional distribution of the number of particles in a decomposable critical branching process $\mathbf{Z}% (m)=(Z_{1}(m),...,Z_{N}(m)),$ with $N$ types of particles at moment $m=n-k,\, k=o(n),$ is investigated given that the extinction moment of the process is $n$.