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2013 Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs
Raphael Lachieze-Rey, Giovanni Peccati
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Electron. J. Probab. 18: 1-32 (2013). DOI: 10.1214/EJP.v18-2104

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.

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Raphael Lachieze-Rey. Giovanni Peccati. "Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs." Electron. J. Probab. 18 1 - 32, 2013. https://doi.org/10.1214/EJP.v18-2104

Information

Accepted: 5 March 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1295.60015
MathSciNet: MR3035760
Digital Object Identifier: 10.1214/EJP.v18-2104

Subjects:
Primary: 60H07
Secondary: 60D05, 60F05, 60G55

JOURNAL ARTICLE
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