## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 47, Number 2 (2015), 530-544.

### Skeletons of near-critical Bienaymé-Galton-Watson branching processes

Serik Sagitov and Maria Conceição Serra

#### Abstract

Skeletons of branching processes are defined as trees of lineages characterized by an appropriate signature of future reproduction success. In the supercritical case a natural choice is to look for the lineages that survive forever (O'Connell (1993)). In the critical case it was suggested that the particles with the total number of descendants exceeding a certain threshold could be distinguished (see Sagitov (1997)). These two definitions lead to asymptotic representations of the skeletons as either pure birth process (in the slightly supercritical case) or critical birth-death processes (in the critical case conditioned on the total number of particles exceeding a high threshold value). The limit skeletons reveal typical survival scenarios for the underlying branching processes. In this paper we consider near-critical Bienaymé-Galton-Watson processes and define their skeletons using marking of particles. If marking is rare, such skeletons are approximated by birth and death processes, which can be subcritical, critical, or supercritical. We obtain the limit skeleton for a sequential mutation model (Sagitov and Serra (2009)) and compute the density distribution function for the time to escape from extinction.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 47, Number 2 (2015), 530-544.

**Dates**

First available in Project Euclid: 25 June 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1435236986

**Digital Object Identifier**

doi:10.1239/aap/1435236986

**Mathematical Reviews number (MathSciNet)**

MR3360388

**Zentralblatt MATH identifier**

1320.60146

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Secondary: 92D25: Population dynamics (general)

**Keywords**

Bienaymé-Galton-Watson process birth and death process decomposable multitype branching process escape from extinction

#### Citation

Sagitov, Serik; Serra, Maria Conceição. Skeletons of near-critical Bienaymé-Galton-Watson branching processes. Adv. in Appl. Probab. 47 (2015), no. 2, 530--544. doi:10.1239/aap/1435236986. https://projecteuclid.org/euclid.aap/1435236986