Limit theorems for critical branching processes in a finite state space Markovian environment

TL;DR

This paper studies the asymptotic behavior of critical branching processes in finite-state Markovian environments, establishing limit theorems for normalized population size and related stochastic processes.

Contribution

It introduces new limit theorems for critical branching processes in Markov environments, including joint laws and Yaglom-type results, extending classical theory to Markovian settings.

Findings

Non-degeneracy of the limit law of Z_n/e^{S_n} conditioned on survival

Joint convergence of S_n/√n and the environment state X_n

Yaglom-type theorem for the joint law of log Z_n and X_n given Z_n>0

Abstract

Let $(Z_n)_{n\geq 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n)_{n\geq 0}$ with values in a finite state space $\mathbb X$. Let $ S_n = \sum_{k=1}^n \ln f_{X_k}'(1)$ be the Markov walk associated to $(X_n)_{n\geq 0}$, where $f_i$ is the offspring generating function when the environment is $i \in \mathbb X$. Conditioned on the event $\{ Z_n>0\}$, we show the non degeneracy of limit law of the normalized number of particles ${Z_n}/{e^{S_n}}$ and determine the limit of the law of $\frac{S_n}{\sqrt{n}} $ jointly with $X_n$. Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of $ \log Z_n$ and $X_n$ given $Z_n>0$.