Vladimir Vatutin

Limit Theorems for Weakly Subcritical Branching Processes in Random Environment

For a branching process in random environment, it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, “supercritical.” This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on nonextinction. Also a functional limit theorem is proved, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.

ON DISTRIBUTION TAILS AND EXPECTATIONS OF MAXIMA IN CRITICAL BRANCHING PROCESSES

We derive the limit behaviour of the distribution tail of the global maximum of a critical Galton-Watson process and also of the expectations of partial maxima of the process, when the offspring law belongs to the domain of attraction of a stable law. Thus the Lindvall (1976) and Athreya (1988) results are extended to the infinite variance case. It is shown that in the general case these two asymptotics are closely related to each other, and the latter follows readily from the former. We also discuss a related problem from the theory of general branching processes.

Limit Theorem for an Intermediate Subcritical Branching Process in a Random Environment

The asymptotic behavior of the survival probability of an intermediate subcritical branching process

Reduced critical branching processes in random environment

Z_n

Stochasticity in the adaptive dynamics of evolution: the bare bones

in a random environment is found when a transformation of the reproduction law of the offspring number is attracted to a stable law

Yaglom Type Limit Theorem for Branching Processes in Random Environment

\alpha\in (1,2]

Weak Convergence of Finite-Dimensional Distributions of the Number of Empty Boxes in the Bernoulli Sieve

. It is shown that the distribution of the random variable

Two-Dimensional Limit Theorem for a Critical Catalytic Branching Random Walk

\{Z_n\}

Catalytic branching random walks in ℤd with branching at the origin

given

Evolution of branching processes in a random environment

Z_n>0