Götz Kersting

Criticality for branching processes in random environment

We study branching processes in an i.i.d. random environment, where the associated random walk is of the oscillating type. This class of processes generalizes the classical notion of criticality. The main properties of such branching processes are developed under a general assumption, known as Spitzers condition in fluctuation theory of random walks, and some additional moment condition. We determine the exact asymptotic behavior of the survival probability and prove conditional functional limit theorems for the generation size process and the associated random walk. The results rely on a stimulating interplay between branching process theory and fluctuation theory of random walks.

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.

A Random Walk Approach to Galton–Watson Trees

There are several constructions connecting random walks to branching trees. Here we discuss an approach linking Galton–Watson trees with arbitrary offspring distribution to random walk excursions resp. bridges. In special situations this leads to a connection to three basic statistics from statistical mechanics. Other applications include the description of random subtrees and the contour process of a Galton–Watson tree.

The Total External Branch Length of Beta-Coalescents

For

The internal branch lengths of the Kingman coalescent

1<\alpha <2

The total external length of the evolving Kingman coalescent

we derive the asymptotic distribution of the total length of {\em external} branches of a Beta

Simulations and a conditional limit theorem for intermediately subcritical branching processes in random environment

(2-\alpha, \alpha)

Path decompositions for Markov chains

-coalescent as the number

The size of the last merger and time reversal in

n

\Lambda

of leaves becomes large. It turns out the fluctuations of the external branch length follow those of