Electronic Journal of Probability

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

Raphael Lachieze-Rey and Giovanni Peccati

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60F05: Central limit and other weak theorems 60G55: Point processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

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

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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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