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

By Raphaël Lachièze-Rey, Giovanni Peccati. 2011

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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.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b7156c6f93ef180caefae4b889a3bdb623de5832

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Citations in the corpus

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals (2012, methodology): “Univariate central limit theorems with bounds on theWasserstein distance for Poisson functionals with finite Wiener–Itô chaos expansions and, in particular, Poisson U statistics are derived in [8], [9], and [16] using a general result from [14].”
Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry (2014, background): “12(iii) in [26] is a qualitative version of Corollary 7.”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “Other relevant contributions in this area are: [16, 49] for explicit bounds in the Kolmogorov distance; [18] for applications to additive functionals of Boolean models; [7, 14, 23, 24, 28] for applications to geometric U -statistics and other non-linear functionals; [50] for CLTs in the framework of”
Central limit theorems for $U$-statistics of Poisson point processes (2011, background): “For graph-theoretical properties of G(Pλ, δ) we refer to [30] and to the more recent developments [8, 17, 21, 23].”
Normal Approximation of Poisson Functionals in Kolmogorov Distance (2012, methodology): “In [4, 5] and [14], Lachi`eze-Rey and Peccati and Reitzner and Schulte used the bound (1) for the Wasserstein distance to derive central limit theorems with explicit rates of convergence for such Poisson functionals occurring in stochastic geometry and random graph theory.”