On the Convergence of Sequences of Branching Processes

Grimvall, Anders

doi:10.1214/aop/1176996496

Cited by 80 publications

(81 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This model arises as the limit of Galton-Watson processes; where individuals behave independently one from each other and each individual gives birth to a random number of offspring, with the same offspring distribution (see for instance Grimvall [19]). More precisely, a CB-process is a [0, ∞]-valued strong Markov process Y = (Y t , t ≥ 0) with cádlág paths such that satisfies the branching property: for all θ ≥ 0 and x, y ≥ 0, E x+y e −θYt = E x e −θYt E y e −θYt .…”

Section: Introductionmentioning

confidence: 99%

Branching Processes in a Lévy Random Environment

Palau

Pardo

2017

Acta Appl Math

View full text Add to dashboard Cite

In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813-857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1-13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.

show abstract

Section: Introductionmentioning

confidence: 99%

Branching Processes in a Lévy Random Environment

Palau

Pardo

2017

Acta Appl Math

View full text Add to dashboard Cite

show abstract

“…In this case, the processes (X (n) k ) k∈Z + cannot start from 0, and the process X (n) nt can be centred by subtracting the initial value X (n) 0 , as in [13]. With suitable normalization, the limiting process will be a Wiener process, and its drift and variance depend on the limiting behaviour of the offspring mean and variance, respectively; see [4,Theorem 4.4]. In our case, the (deterministic) time change concerning the limit process ( M(t)) t∈R + is usually not linear, which is an effect of the immigration part.…”

Section: Introductionmentioning

confidence: 99%

Fluctuation limit of branching processes with immigration and estimation of the means

Ispány¹,

Pap²,

Zuijlen³

2005

Adv. Appl. Probab.

View full text Add to dashboard Cite

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2 , in contrast to both the subcritical case, in which it is n 1/2 , and the nearly critical case with positive limiting offspring variance, in which it is n.

show abstract

“…Cette approche a été généralisée, notamment au cas avec immigration [15], mais pose de sérieuses difficultés dans le cas d'environnements fluctuants. Dans la même veine, Grimwall [31] donne une condition nécessaire et suffisante pour la convergence de processus de Galton Watson renormalisés, grâce à des tableaux triangulaires. Ces convergences peuvent être reliées à la convergence d'un triplet caractéristique de la loi de reproduction du processus de Galton-Watson.…”

Section: Processus De Branchement à Temps Et Espace D'états Continusunclassified

Probabilités et biologie

et al. 2014

View full text Add to dashboard Cite

Abstract. This session presents various aspects of stochastic populations models, mainly focused on branching mechanism: branching process for random size population in random environment; coexistence in branching process for constant size population; study of the external branch lengths for genealogical tree in a constant size population; model for the size of cancer tumor (random size population) with radiotherapy treatment.Résumé. Cette session présente divers aspects de la modélisation probabiliste des populations, en mettant l'accent sur le branchement : processus de branchement à taille de population aléatoire en milieux aléatoires ; coexistence de la diversité dans des processus de branchement à taille de population constante ; étude des longueurs de branches externes dans les arbres généalogiques de population à taille constante ; modélisation de la taille d'une tumeur (population à taille aléatoire) traitée par radiothérapie.

show abstract

On the Convergence of Sequences of Branching Processes

Cited by 80 publications

References 0 publications

Branching Processes in a Lévy Random Environment

Branching Processes in a Lévy Random Environment

Fluctuation limit of branching processes with immigration and estimation of the means

Probabilités et biologie

Contact Info

Product

Resources

About