zbMATH — the first resource for mathematics

Reduced subcritical Galton-Watson processes in a random environment. (English) Zbl 0938.60090
The authors study the structure of a reduced subcritical BGW process in a random environment given by a sequence $$(f_n)$$ of i.i.d. offspring generating functions with probability law $$\mathbb{P}$$. The quantity considered is the number of particles in generation $$m$$ having nonempty offspring in generation $$n$$. The behaviour is essentially different depending on whether $$\mathbb{E} f_0'(1)\log f_0'(1)$$ is finite non-positive or finite positive. In the first case, the most recent common ancestor of the non-empty $$n$$th generation is ‘located’ close to the moment $$n$$, i.e., the situation is similar to that for classical subcritical BGW processes. In the second case, however, a new hybrid type of behaviour occurs: The most recent common ancestor is located exactly at the beginning of the genealogical tree, just as in classical supercritical BGW processes. This implies the so-called “branchless thick trunk” phenomenon. Relations to random walks in random environment are also discussed.

MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G50 Sums of independent random variables; random walks
Full Text: