Galton--Watson Branching Processes in a Random Environment I: Limit Theorems

How Many Families Survive for a Long Time?

“…Since p ≪ n, we deduce by (15), (12) and properties of regularly varying functions that if n → ∞ then for any ε > 0…”

“…where we have used (15) to justify the last inequality. Since f 0,t (0) → f 0,∞ (0) ∈ (0, 1) P + -a.s. as t → ∞, letting in (29) first n → ∞ and then l → ∞, we get in account of (22) …”

“…It follows from (8) that V and U give rise to probability measures P + x and P − x being important for subsequent arguments. The construction procedures of these probability measures are explained for P + x in [1] and for P − x in [2] and [15] in detail. We only recall that if T 0 , T 1 , .…”

Reduced multitype critical branching processes in random environment

“…random environment was found in [16] and [32] for the single-type processes under the assumption that the second moment of the increments of the associated random walk is finite. In subsequent papers [13], [31], [18]- [20] the asymptotics of the survival probability and the behavior of the distribution of the number of particles in the process for an essentially wider class of single-type critical BPREs were investigated.…”

Critical Branching Process with Two Types of Particles Evolving in Asynchronous Random Environments

“…Theorem 2. Under Conditions A1-A3 as n → ∞ (9) ζ (1) n , ζ (2) n d → ζ (1) , ζ (2) , Downloaded 11/19/14 to 129.49.23.145. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php where the random variables ζ (i) are independent identically distributed and take values in (0, 1] with probability 1.…”