# $U$ -Statistics of Ornstein–Uhlenbeck Branching Particle System

Journal of Theoretical Probability, Sep 2014

We consider a branching particle system consisting of particles moving according to the Ornstein–Uhlenbeck process in $\mathbb {R}^d$ and undergoing a binary, supercritical branching with a constant rate $\lambda >0$. This system is known to fulfill a law of large numbers (under exponential scaling). Recently the question of the corresponding central limit theorem (CLT) has been addressed. It turns out that the normalization and the 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 Ornstein–Uhlenbeck process. In the present paper, we extend those results to $U$-statistics of the system, proving a law of large numbers and CLT.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs10959-013-0503-2.pdf

Radosław Adamczak, Piotr Miłoś. $U$ -Statistics of Ornstein–Uhlenbeck Branching Particle System, Journal of Theoretical Probability, 2014, 1071-1111, DOI: 10.1007/s10959-013-0503-2