Branching processes exhibit a particularly rich longtime behaviour when evolving in a random environment. Then the transition from subcriticality to supercriticality proceeds in several steps, and there occurs a second ‘transition’ in the subcritical phase (besides the phase-transition from (sub)criticality to supercriticality). Here we present and discuss limit laws for branching processes in critical and subcritical i.i.d. environment. The results rely on a stimulating interplay between branching process theory and random walk theory. We also consider a spatial version of branching processes in random environment for which we derive extinction and ultimate survival criteria.

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 Spitzer's 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.

Let T be a critical or subcritical Galton‐Watson family tree with possibly infinite variance. We are interested in the shape of T conditioned to have a large total number of vertices. For this purpose we study random trees whose conditional distribution given their size is the same as the respective conditional distribution of T. These random family trees have a simple probabilistic structure if decomposed along the lines of descent of a number of distinguished vertices chosen uniformly at random. The shape of the subtrees spanned by the selected vertices and the root depends essentially on the tail of the offspring distribution: While in the finite variance case the subtrees are asymptotically binary, other shapes do persist in the limit if the variance is infinite. In fact, we show that these subtrees are Galton‐Watson trees conditioned on their total number of leaves. The rescaled total size of the trees is shown to have a gamma limit law. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004

Let (Zn)n≥0 be a critical Galton-Watson branching process with finite variance. We give a new proof of a classical result by Yaglom that Z n conditioned on Zn > 0 has an exponential lirnit law.

Consider a critical binary continuous-time Galton-Watson tree size-biased according to the number of particles at time $t$. Decompose the population at $t$ according to the particles' degree of relationship with a distinguished particle picked purely at random from those alive at $t$. Keeping track of the times when the different families grow out of the distinguished line of descent and the related family sizes at $t$, we represent this relationship structure as a point process in a time-size plane. We study limits of these point processes in the single- and some multitype case.

We give a probabilistic construction of the Galton--Watson treeconditioned on its height by decomposing it along the line of descentto the left--most particle at maximal height. This construction providesa representation of the final generation size as a sum of independentincrements. Based on this representation we prove limit laws for thefinal generation size of a Galton--Watson tree conditioned on its heightand total progeny, respectively.1 IntroductionLet T denote the random family...

Random splitting trees share the striking independence properties of the continuous time binary Galton-Watson tree. They can be represented by Poisson point processes and their contour processes are strong Markov processes. Here we study splitting trees conditioned on extinction, respectively non-extinction as well as size-biased splitting trees. We give explicit probabilistic constructions of those trees by decomposing them into independent parts along a distinguished line of descent. The size-biased trees are shown to have stationary contour processes. Splitting trees are related to M/G/1-queuing systems which allows to translate the results on the trees into statements on the queues.