Comparison Results for Branching Processes in Random Environments

Abstract

In this note we consider branching processes whose behavior depends on a dynamic random environment, in the sense that we assume that the offspring distributions of individuals are parametrized, over time, by the realizations of a process describing the environmental evolution. We study how the variability in time of the environment modifies the variability of total population by considering two branching processes of this kind (but subjected to different environments). We also provide conditions on the random environments in order to stochastically compare their marginal distributions in the increasing convex sense. Weaker conditions are also provided for comparisons at every fixed time of the expected values of the two populations.

Convex Comparisons for Random Sums in Random Environments and Applications

“…From the previous result, we can easily get the following comparison result for two branching processes defined on two different random environments (see Pellerey [30] for further details)). COROLLARY 4.2: Consider the stochastic processes Z(θ) = {Z n (θ 0 , .…”

“…Several applications involve medicine, molecular and cellular biology, human evolution, physics or actuarial science (see Rolski, Schmidli, Schmidt, and Teugeis [32], Ross [33], or Kimmel and Axelrod [16]). In this subsection, we provide a result dealing with stochastic comparisons between two branching processes defined on random environments, which is closely related to Theorem 2.2 in Pellerey [30]. The branching processes on random environments that we consider here are defined as follows.…”