Branching in a Markovian Environment
2023, v.29, Issue 1, 1-33
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 environment $j$ given that the initial environment 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.
Keywords: extinction matrix, martingale central limit theorem, matrix generating function, stochastic abelian network