On Asymptotic Expansion in the Random Allocation of Particles by Sets

We consider a scheme of equiprobable allocation of particles into cells by sets. The Edgeworth type asymptotic expansion in the local central limit theorem for a number of empty cells left after allocation of all sets of particles is derived.Comment: 15 page

Reduced two-type decomposable critical branching processes with possibly infinite variance

International audienc

Branching Processes: Variation, Growth, and Extinction of Populations

Biology takes a special place among the other natural sciences because biological units, be they pieces of DNA, cells, or organisms, reproduce more or less faithfully. Like any other biological process, reproduction has a large random component. The theory of branching processes was developed especially as a mathematical counterpart to this most fundamental of biological processes. This active and rich research area allows us to determine extinction risks and predict the development of population composition, and also uncover aspects of a population's history from its current genetic composition. Branching processes play an increasingly important role in models of genetics, molecular biology, microbiology, ecology, and evolutionary theory. This book presents this body of mathematical ideas for a biological audience, but should also be enjoyable to mathematicians, if only for its rich stock of rich biological examples. It can be read by anyone with a basic command of calculus, matrix algebra, and probability theory. More advanced results from basic probability theory are treated in a special appendix

A Decomposable Branching Process in a Markovian Environment

A population has two types of individuals, with each occupying an island. One of those, where individuals of type 1 live, offers a variable environment. Type 2 individuals dwell on the other island, in a constant environment. Only one-way migration is possible. We study then asymptotics of the survival probability in critical and subcritical cases

Functional limit theorems for strongly subcritical branching processes in random environment

For a strongly subcritical branching process (Zn)n[greater-or-equal, slanted]0 in random environment the non-extinction probability at generation n decays at the same exponential rate as the expected generation size and given non-extinction at n the conditional distribution of Zn has a weak limit. Here we prove conditional functional limit theorems for the generation size process (Zk)0[less-than-or-equals, slant]k[less-than-or-equals, slant]n as well as for the random environment. We show that given the population survives up to generation n the environmental sequence still evolves in an i.i.d. fashion and that the conditioned generation size process converges in distribution to a positive recurrent Markov chain.Branching process Random environment Random walk Change of measure Positive recurrent Markov chain Functional limit theorem