2021/06/21 by Lila Greco, Lionel Levine, Greco, Lila +1
Business, Management and Accounting · Mathematics · #15A24 #15B51 #60F05 #60J10 #60J80 #60K37 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2106.11249
openalex publication_date 2021/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.