Central limit theorems for branching processes under mild assumptions on the mean semigroup
Bertrand Cloez, Nicolás Zalduendo
TL;DR
This work proves central limit theorems for a broad class of supercritical branching Markov processes in infinite dimensions under mild mean-semigroup ergodicity and a fourth-moment condition. The authors use Stein's method to obtain Gaussian fluctuations around the LLN limit and derive explicit convergence rates, applicable across small, critical, and many infinite-dimensional settings, including non-symmetric and non-diffusive dynamics. The results recover known small- and critical-branching regimes and extend to a wide family of processes such as age-structured and growth-fragmentation models, with the house-of-cards model illustrating non-smooth mean semigroups. Overall, the paper provides a unifying framework for Gaussian fluctuations in branching systems with explicit variance structures and rates, enhancing theoretical understanding and potential statistical applications.
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.