Central limit theorems for branching processes under mild assumptions on the mean semigroup

TL;DR

This paper proves central limit theorems for a broad class of supercritical branching Markov processes in infinite dimensions, using weak assumptions on the mean semigroup and Stein's method for convergence rates.

Contribution

It extends known CLT regimes to wider processes under mild conditions, without requiring symmetry or detailed spectral information.

Findings

Established CLTs for supercritical branching processes in infinite dimensions.

Provided convergence rates using Stein's method.

Extended CLT regimes to broader classes of processes.

Abstract

We establish central limit theorems for a large class of supercritical branching Markov processes in infinite dimension with spatially dependent and non-necessarily local branching mechanisms. This result relies on a fourth moment assumption and the exponential convergence of the mean semigroup in a weighted total variation norm. This latter assumption is pretty weak and does not necessitate symmetric properties or specific spectral knowledge on this semigroup. In particular, we recover two of the three known regimes (namely the small and critical branching processes) of convergence in known cases, and extend them to a wider family of processes. To prove our central limit theorems, we use the Stein's method, which in addition allows us to newly provide a rate of convergence to this type of convergence.