Reduced subcritical Galton-Watson processes in a random environment. (English) Zbl 0938.60090

Fleischmann, Klaus; Vatutin, Vladimir A.

Adv. Appl. Probab. 31, No. 1, 88-111 (1999).

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.

Reviewer: H.Hering (Göttingen)

MSC:

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

Keywords:

branching process; random environment; reduced process; reduced tree; conditional limit theorem; source time; random walk; ladder epochs