U-statistics of Ornstein-Uhlenbeck branching particle system

Abstract

We consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in $\Rd$ and undergoing a binary, supercritical branching with a constant rate $\lambda>0$. This system is known to fulfil a law of large numbers (under exponential scaling). Recently the question of the corresponding central limit theorem has been addressed. It turns out that the normalization and form of the limit in the CLT fall into three qualitatively different regimes, depending on the relation between the branching intensity and the parameters of the Orstein-Uhlenbeck process. In the present paper we extend those results to $U$-statistics of the system proving a law of large numbers and a central limit theorem.