Critical multitype branching processes in a random environment

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.…”

“…We will use the following statement which follows in an obvious way from the proof of Theorem 1 in [29] and formula (3.4) in the mentioned paper. …”

“…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…”