Asymptotic properties of multitype critical branching processes evolving in a random environment

Abstract

For an extended class of multitype critical branching processes in a random environment, the asymptotic behaviour of the survival probability is found under the conditions which are weaker than those known earlier even for the single-type case. A functional limit theorem is proved for the number of particles in the process at moments nt, 0 t 1, conditioned on its survival up to moment n.

Reduced multitype critical branching processes in random environment

“…In [25] - [29] a number of theorems were established for an extended class of critical multitype branching processes in random environment which describe the asymptotic of the survival probability and the limit behavior of the number of particles in the process conditioned on the survival of the process up to a distant moment.…”

“…The sequence S = ( 0 , 1 , ...) is called (see [25], [29]) the associated random walk for the multitype branching process {Z , ≥ 0} in random environment .…”

“…It is known that many important properties of single-type and multitype branching processes in random environment are closely connected with the properties of the associated random walks (see, for instance, [13], [30], [25], [29]). Following [25] and [29], we call a -type branching process {Z , ≥ 0} satisfying Assumption A1 in random environment critical if its associated random walk is of the oscillating type, i.e., if lim sup →∞ = +∞ a.s. and lim inf →∞ = −∞ a.s. Clearly, if Assumption B3 is valid, then the process under consideration is critical.…”

“…Following [25] and [29], we call a -type branching process {Z , ≥ 0} satisfying Assumption A1 in random environment critical if its associated random walk is of the oscillating type, i.e., if lim sup →∞ = +∞ a.s. and lim inf →∞ = −∞ a.s. Clearly, if Assumption B3 is valid, then the process under consideration is critical. Let…”

Multitype branching processes evolving in a Markovian environment

“…Representatives of this class allow for the possibility EX 2 D 1 and even do not require the existence of the expectation of X . Papers [16] and [18] consider the processes from an extended class of multitype critical branching processes generated by a sequence of independent and identically distributed random variables and such whose associated random walk satisfy the Spitzer-Doney condition (see Condition C1 below) and whose mean matrices of the reproduction laws of all generations have one and the same left (or right) eigenvector corresponding to their Perron roots.…”