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

Raphael Lachieze-Rey (1, 2) , Giovanni Peccati (2)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 FSTC - Faculté des Sciences, de la Technologie et de la Communication

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 sequences of U-statistics with rescaled kernels.

Mots clés: Central Limit Theorems; Contractions; Malliavin Calculus; Poisson Limit Theorems; Poisson Space; Random Graphs; Stein's Method; U-statistics; Wasserstein Distance; Wiener Chaos

Domaines: Probabilités [math.PR]

Raphael Lachieze-Rey, Giovanni Peccati. Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs.. 2011. ⟨hal-00646866v2⟩