## Electronic Journal of Probability

### 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.

#### Article information

Source
Electron. J. Probab. Volume 18 (2013), paper no. 32, 32 pp.

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

https://projecteuclid.org/euclid.ejp/1465064257

Digital Object Identifier
doi:10.1214/EJP.v18-2104

Mathematical Reviews number (MathSciNet)
MR3035760

Zentralblatt MATH identifier
1295.60015

Rights

#### Citation

Lachieze-Rey, Raphael; Peccati, Giovanni. Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs. Electron. J. Probab. 18 (2013), paper no. 32, 32 pp. doi:10.1214/EJP.v18-2104. https://projecteuclid.org/euclid.ejp/1465064257

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