Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Small positive values for supercritical branching processes in random environment

Vincent Bansaye and Christian Böinghoff

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Abstract

Branching Processes in Random Environment (BPREs) $(Z_{n}\colon\ n\geq0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times.

We describe the asymptotic behavior of $\mathbb{P}(1\leq Z_{n}\leq k\vert Z_{0}=i)$, $k,i\in\mathbb{N}$ as $n\rightarrow\infty$. More precisely, we characterize the exponential decrease of $\mathbb{P}(Z_{n}=k\vert Z_{0}=i)$ using a spine representation due to Geiger. We then provide some bounds for this rate of decrease.

If the reproduction laws are linear fractional, this rate becomes more explicit and two regimes appear. Moreover, we show that these regimes affect the asymptotic behavior of the most recent common ancestor, when the population is conditioned to be small but positive for large times.

Résumé

Les processus de branchement en environnement aléatoire $(Z_{n}\colon\ n\geq0)$ sont une généralisation des processus de Galton Watson où à chaque génération, la reproduction est choisie de manière i.i.d. Dans le régime surcritique, ces processus survivent avec probabilité positive et croissent alors géométriquement. Ce papier considère l’événement rare où le processus prend des valeurs non nulles mais bornées en temps long.

Nous décrivons ainsi le comportement asymptotique de $P(1\leq Z_{n}\leq k\vert Z_{0}=i)$ quand $n\rightarrow\infty$. Plus précisément, nous caractérisons la vitesse exponentielle àlaquelle $\mathbb{P}(Z_{n}=k\vert Z_{0}=i)$ tend vers zéro en utilisant une représentation en épine due à Geiger. Nous donnons alors des bornes pour cette vitesse.

Si la loi de reproduction est linéaire fractionnaire, la vitesse devient plus explicite et deux régimes apparaissent. Nous montrons par ailleurs que ces régimes affectent le comportement asymptotique de l’ancêtre commun le plus récent de la population en vie à l’instant n quand cette dernière est conditionnée à prendre de petites valeurs en temps long.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 3 (2014), 770-805.

Dates
First available in Project Euclid: 20 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1403276998

Digital Object Identifier
doi:10.1214/13-AIHP538

Mathematical Reviews number (MathSciNet)
MR3224289

Zentralblatt MATH identifier
1307.60114

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments 60J05: Discrete-time Markov processes on general state spaces 60F17: Functional limit theorems; invariance principles 92D25: Population dynamics (general)

Keywords
Supercritical branching processes Random environment Large deviations Phase transitions

Citation

Bansaye, Vincent; Böinghoff, Christian. Small positive values for supercritical branching processes in random environment. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 3, 770--805. doi:10.1214/13-AIHP538. https://projecteuclid.org/euclid.aihp/1403276998


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