Séminaire de Probabilités

Asymptotic fluctuations of supercritical general branching processes

by Matthias Meiners

Amphi Schwartz (IMT)

Amphi Schwartz



The Crump-Mode-Jagers (CMJ) process is a fairly general branching process
that unifies and extends earlier models of individual-based branching processes.
Nerman's celebrated law of large numbers (1981) states that,
for a supercritical CMJ process $(\mathcal{Z}_t)_{t \geq 0}$,
under some mild assumptions, $e^{-\alpha t} \mathcal Z_t$
converges almost surely as $t \to \infty$ to a random variable $aW$.
Here, $\alpha>0$ is the Malthusian parameter,
$a$ is a constant and $W$ is the limit of Nerman's martingale,
which is positive on the event that the population survives.

I shall present a recently obtained central limit theorem for the CMJ process
that explains how $\mathcal Z_t$ fluctuates around its first-order term $e^{\alpha t} a W$.

The talk is based on joint work with Alexander Iksanov (Kyiv) and Konrad Kolesko (Gie\ss en).