[Paper Review] Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometric U-statistics

Raphaël Lachièze-Rey, Giovanni Peccati|arXiv (Cornell University)|May 3, 2012

Point processes and geometric inequalities26 references4 citations

TL;DR

This paper establishes central limit theorems for U-statistics with rescaled kernels on marked Poisson processes, using contraction operators to derive explicit Wasserstein distance bounds. It proves that geometric U-statistics—such as connected k-tuples in random networks with random radii—converge to a normal distribution at rate $\lambda^{-1/2}$ under integrability conditions on the tail of the radius distribution.

ABSTRACT

Continuing the analysis initiated in Lachiéze-Rey and Peccati (2011), we use contraction operators to study the normal approximation of random variables having the form of a U-statistic written on the points in the support of a random Poisson measure. Applications are provided: to boolean models, and coverage of random networks.

Motivation & Objective

• To extend the contraction-based normal approximation framework from [10] to U-statistics with rescaled kernels on marked Poisson processes.
• To characterize the asymptotic normality of geometric U-statistics arising in stochastic geometry, such as connected subgraphs in random networks.
• To provide explicit, quantitative bounds on the Wasserstein distance between the distribution of such U-statistics and a Gaussian limit.
• To handle edge effects in spatial processes by replacing toroidal metrics with Euclidean distances and incorporating random interaction radii.
• To complete the characterization of geometric U-statistics initiated in Reitzner and Schulte (2011), using Hoeffding decompositions and contraction-based bounds.

Proposed method

• Uses contraction operators to bound the Wasserstein distance between a U-statistic and a centered Gaussian distribution.
• Applies the theory of finite Wiener–Itô chaos expansions and Hoeffding decompositions for symmetric U-statistics.
• Employs rescaled kernels and stationary assumptions to analyze fluctuations in spatial point processes.
• Derives sufficient conditions via integral estimates involving the tail function $F(r) = \mathbb{P}(R_1 > r)$ of the random radius distribution.
• Performs spherical changes of variables to bound integrals over $\mathbb{R}^{d(k-1)}$, reducing the problem to moment conditions on $F(r)$.
• Establishes convergence rates by verifying integrability of $\int_{\mathbb{R}_+} F(r) r^{4d-1+\varepsilon} dr < \infty$ for some $\varepsilon > 0$.

Experimental results

Research questions

• RQ1 Under what conditions does a U-statistic with a rescaled kernel on a marked Poisson process converge to a normal distribution?
• RQ2 How can contraction operators be used to derive explicit bounds on the Wasserstein distance for such U-statistics?
• RQ3 What are the sufficient moment conditions on the radius distribution that ensure asymptotic normality in random network coverage models?
• RQ4 How do edge effects in spatial processes affect the limiting behavior of geometric U-statistics, and how can they be mitigated?
• RQ5 Can the framework be extended to characterize joint convergence of multiple chaotic components in U-statistics?

Key findings

• For $k \geq 2$, the number $N_{k,\lambda}$ of $k$-tuples of devices in a random network with random radii converges to a normal distribution as $\lambda \to \infty$.
• The variance of $N_{k,\lambda}$ grows linearly with $\lambda$, satisfying $\mathrm{Var}(N_{k,\lambda}) \sim c_k \lambda$ for some $c_k > 0$.
• The Wasserstein distance between the centered and normalized $N_{k,\lambda}$ and a standard Gaussian random variable is bounded by $C_k \lambda^{-1/2}$.
• A sufficient condition for this convergence is $\int_{\mathbb{R}_+} F(r) r^{4d-1+\varepsilon} dr < \infty$ for some $\varepsilon > 0$, where $F(r)$ is the tail of the radius distribution.
• The result holds even when using Euclidean distances instead of toroidal metrics, thus removing edge effects.
• The contraction-based method yields bounds that are simpler and more effective than diagram-based approaches, especially for noncircular partitions.