Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs. - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Electronic Journal of Probability Année : 2013

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

Résumé

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.
Fichier principal
Vignette du fichier
LR_P_ejp_2.pdf (712.27 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00646866 , version 1 (30-11-2011)
hal-00646866 , version 2 (14-02-2012)
hal-00646866 , version 3 (23-06-2012)

Identifiants

Citer

Raphaël Lachièze-Rey, Giovanni Peccati. Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs.. Electronic Journal of Probability, 2013, 18, pp.32. ⟨10.1214/EJP.v18-2104⟩. ⟨hal-00646866v3⟩
180 Consultations
166 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More