CLT for U-statistics of Ornstein-Uhlenbeck branching particle system with small branching rate

TL;DR

This paper establishes a central limit theorem for U-statistics of an Ornstein-Uhlenbeck branching particle system with small growth rate, extending understanding of fluctuations in such stochastic systems.

Contribution

It proves a CLT for U-statistics in the system and analyzes how the growth rate influences the second order behavior, which is a novel extension.

Findings

Proved CLT for the particle system's U-statistics.

Derived limit distributions in terms of multiple stochastic integrals.

Analyzed the impact of small growth rate on system fluctuations.

Abstract

In this paper we consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in R^d 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). In the paper we prove the corresponding central limit theorem. Moreover, in the second part of the paper we consider U-statistics of the system, for which, under mild assumptions, we prove a law of large numbers and a central limit theorem. The limits are expressed in terms of multiple stochastic integrals with respect to a random Gaussian measure. The second order behaviour depends qualitatively on the growth rate of the system. In this paper we concentrate on the case when the growth rate is relatively small comparing to smoothing properties of particles' movement.