Properties of multitype subcritical branching processes in random environment

TL;DR

This paper analyzes the long-term behavior of multitype subcritical branching processes in random environments, revealing how survival probabilities decay exponentially and how the distribution of particles stabilizes over time.

Contribution

It establishes the asymptotic form of survival probabilities and shows the independence of the limiting distribution from initial conditions in multitype subcritical processes.

Findings

Survival probability decays as $C(z)\lambda^n$ for large $n$

Limiting distribution of particles is independent of initial vector $z$

Explicit expression for the constant $C(z)$ and Lyapunov exponent $\\lambda$

Abstract

We study properties of a $p-$type subcritical branching process in random environment initiated at moment zero by a vector $\mathbf{z}=\left( z_{1},..,z_{p}\right) $\ of particles of different types. Assuming that the process belongs to the class of the so-called strongly subcritical processes we show that its survival probability to moment $n$\ behaves for large $n$\ as $C(\mathbf{z})\lambda ^{n}$\ where $\lambda $\ is the upper Lyapunov exponent for the product of mean matrices of the process and $C(\mathbf{z})$% \ is an explicitly given constant. We also demonstrate that the limiting conditional distribution of the number of particles given the survival of the process for a long time does not depend on the vector $\mathbf{z}$ of the number of particles initiated the process.