Comparison Results for Branching Processes in Random Environments

We consider branching processes whose behaviors depend on a dynamic random environment, in the sense that we assume the offsprings distributions of individuals parametrized, during 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: considered two branching processes of such kind, but subjected to different environments, we provide conditions on the random environments in order to stochastically compare their marginal distributions in increasing convex sense. Weaker conditions are also provided for comparisons at every fixed time of the expected values of the two populations.