Asymptotics of Continuous-Time Discrete State Space Branching Processes for Large Initial State

M. Mohle, B. Vetter

2021, v.27, Issue 1, 1-42

ABSTRACT

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\tire 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.

Keywords: branching process; generalized Mehler semigroup; Neveu's continuous-state branching process; Ornstein\tire Uhlenbeck type process; self-decomposability; stable law; time-inhomogeneous process; weak convergence

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