Branching in a Markovian Environment

TL;DR

This paper studies a branching process influenced by a Markovian environment, establishing conditions for survival, and deriving laws of large numbers, central limit theorems, and extinction probabilities.

Contribution

It introduces a first moment criterion for survival, and develops a matrix generating function approach to analyze extinction probabilities in Markovian environments.

Findings

Survival depends on a specific first moment condition.

Law of large numbers and CLT are proved for population size.

Extinction probabilities are characterized via a fixed point of a matrix generating function.

Abstract

A branching process in a Markovian environment consists of an irreducible Markov chain on a set of "environments" together with an offspring distribution for each environment. At each time step the chain transitions to a new random environment, and one individual is replaced by a random number of offspring whose distribution depends on the new environment. We give a first moment condition that determines whether this process survives forever with positive probability. On the event of survival we prove a law of large numbers and a central limit theorem for the population size. We also define a matrix-valued generating function for which the extinction matrix (whose entries are the probability of extinction in state j given that the initial state is i) is a fixed point, and we prove that iterates of the generating function starting with the zero matrix converge to the extinction matrix.