Limit theorems for continuous-state branching processes with immigration - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Advances in Applied Probability Année : 2022

Limit theorems for continuous-state branching processes with immigration

Chunhua Ma
  • Fonction : Auteur
Linglong Yuan
  • Fonction : Auteur

Résumé

Abstract A continuous-state branching process with immigration having branching mechanism $\Psi$ and immigration mechanism $\Phi$ , a CBI $(\Psi,\Phi)$ process for short, may have either of two different asymptotic regimes, depending on whether $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ or $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ . When $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ , the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ , immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ ( Bull. Amer. Math. Soc. 78 , 1972).

Dates et versions

hal-03860200 , version 1 (18-11-2022)

Identifiants

Citer

Clément Foucart, Chunhua Ma, Linglong Yuan. Limit theorems for continuous-state branching processes with immigration. Advances in Applied Probability, 2022, 54 (2), pp.599-624. ⟨10.1017/apr.2021.43⟩. ⟨hal-03860200⟩
21 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More