Asymptotics of continuous-time discrete state space branching processes for large initial state. (English) Zbl 1480.60264

Möhle, Martin; Vetter, Benedict

Markov Process. Relat. Fields 27, No. 1, 1-42 (2021).

Summary: Scaling limits for continuous-time branching processes with discrete state space are provided as the initial state tends to infinity. Depending on the finiteness or non-finiteness of the mean and/or the variance of the offspring distribution, the limits are in general time-inhomogeneous Gaussian processes, time-inhomogeneous generalized Ornstein-Uhlenbeck type processes or continuous-state branching processes. We also provide transfer results showing how specific asymptotic relations for the probability generating function of the offspring distribution carry over to those of the one-dimensional distributions of the branching process.

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60F05	Central limit and other weak theorems
60F17	Functional limit theorems; invariance principles
60G50	Sums of independent random variables; random walks
60J27	Continuous-time Markov processes on discrete state spaces

Keywords:

branching process; generalized Mehler semigroup; Neveu’s continuous-state branching process; Ornstein-Uhlenbeck type process; self-decomposability; stable law; time-inhomogeneous process; weak convergence