# Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs.

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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Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2013, 18, pp.32. 〈10.1214/EJP.v18-2104〉
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https://hal.archives-ouvertes.fr/hal-00646866
Contributeur : Raphael Lachieze-Rey <>
Soumis le : samedi 23 juin 2012 - 12:57:59
Dernière modification le : jeudi 11 janvier 2018 - 06:19:44
Document(s) archivé(s) le : jeudi 15 décembre 2016 - 18:00:43

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Raphaël Lachièze-Rey, Giovanni Peccati. Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs.. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2013, 18, pp.32. 〈10.1214/EJP.v18-2104〉. 〈hal-00646866v3〉

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