Abstract
We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit theorems. These conditions can always be expressed in terms of contraction operators or, equivalently, fourth cumulants. Our findings are specifically tailored to deal with the normal approximation of the geometric $U$-statistics introduced by Reitzner and Schulte (2011). In particular, we shall provide a new analytic characterization of geometric random graphs whose edge-counting statistics exhibit asymptotic Gaussian fluctuations, and describe a new form of Poisson convergence for stationary random graphs with sparse connections. In a companion paper, the above analysis is extended to general $U$-statistics of marked point processes with possibly rescaled kernels.
Highlights
This paper concerns the normal approximation of random variables living inside a fixed sum of Wiener chaoses associated with a Poisson measure over a Borel measure space
Our main theoretical tools come from the two papers [26, 27], respectively by Peccati et al and Peccati and Zheng, where the normal approximation of functional of Poisson measures is studied by combining two probabilistic techniques, namely the Stein’s method and the Malliavin calculus of variations
We shall focus on conditions implying that a given sequence of random variables satisfies a central limit theorem (CLT), where the convergence in distribution takes place in the sense of the Wasserstein distance
Summary
This paper concerns the normal approximation of random variables living inside a fixed sum of Wiener chaoses associated with a Poisson measure over a Borel measure space. – In Theorem 3.5, we shall prove that conditions for asymptotic normality can be expressed in terms of norms of contraction operators (see Section 2.2) These analytic objects already appear in CLTs living inside a fixed Wiener chaos (see [26, 27]), and are a crucial tool in order to effectively assess bounds based on Malliavin operators. We shall use our results in order to provide an exhaustive characterization of stationary geometric random graphs whose edge counting statistics exhibit asymptotic Gaussian fluctuations (see Theorem 4.11). This family of geometric graphs contains e.g. interval graphs and disk graphs – see e.g. The rest of this section is devoted to the formal presentation of the main problems that are addressed in this paper
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