Small positive values for supercritical branching processes in random environment

Bansaye, Vincent; Böinghoff, Christian

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 770-805.

Les processus de branchement en environnement aléatoire $(Z_n:n\ge 0)$ 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 $P(1\le Z_n\le k|Z_0=i)$ quand $n\to \infty$. Plus précisément, nous caractérisons la vitesse exponentielle àlaquelle $\mathbb{P}(Z_n=k|Z_0=i)$ 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.

Branching Processes in Random Environment (BPREs) $(Z_n:n\ge 0)$ 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 $\mathbb{P}(1\le Z_n\le k|Z_0=i)$, $k,i\in \mathbb{N}$ as $n\to \infty$. More precisely, we characterize the exponential decrease of $\mathbb{P}(Z_n=k|Z_0=i)$ 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.

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