Year: 2021
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Greco, Lila, Levine, Lionel
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3176040268