Multitype branching processes evolving in a Markovian environment

Abstract

Multitype critical branching processes evolving in a Markovian random environment are considered. A conditional limit theorem on the distribution of the number of particles in the process at moment n is proved given survival of the process to this moment.

Branching processes in a Markov random environment

“…In particular, we extend some results of [18] and [19], and generalize some results of [9] and [10] to the case of a Markov random environment with countable set of states.…”

“…To this end we follow the arguments in the proof of Theorem 1 in [19], which proves (37) with = 1. We give the corresponding arguments for completeness of presentation only.…”

“…The asymptotics of the nonextinction probability of a critical branching process with one type of particles in a Markov environment with nite set of states were obtained in [21]. For a critical branching process with ≥ 1 types of particles evolving in a Markov random environment with countable set of states, the asymptotics of the nonextinction probability were found in [18], and later in [19] the limiting behaviour of the number of particles in this process at time as → ∞ was studied.…”

“…We shall also need the following assertion, which is a combination of Corollary 1 of [18] and Lemma 4 of [19].…”

Reduced multitype critical branching processes in random environment

“…By Corollary 1, as → ∞, the uniform convergence (42) where the matrix is the same as in (27). Let be the × matrix whose components are equal to min , where min is defined in (23).…”