Poisson U-Statistics: Foundations & Analysis

• Poisson U-statistics are defined as symmetric kernel sums over all ordered k-tuples from a Poisson point process, enabling finite Wiener–Itô chaos expansions.
• They support rigorous limit theorems, providing explicit central and non-Gaussian limit results with quantitative convergence rates based on contraction norms.
• Applications span from subgraph counts in random geometric graphs to inference procedures on manifolds, underscoring their significance in spatial statistics and stochastic geometry.

A Poisson U-statistic is a random variable or vector defined as the sum of a symmetric kernel evaluated over all ordered $k$-tuples of distinct points in a Poisson point process. The study of Poisson U-statistics incorporates elements from stochastic geometry, Malliavin calculus, concentration of measure, limit theorems, and functional approximation theory. The structure of Poisson U-statistics permits the use of finite Wiener–Itô chaos expansions, enabling central limit theorems, precise rates of convergence, explicit moment/cumulant formulas, and sharp concentration inequalities. Applications are diverse, encompassing subgraph counts in random geometric graphs, intersection volumes in Poisson hyperplane processes, functional statistics on manifolds, and inference procedures in spatial statistics.

1. Definition, Integrability, and Wiener–Itô Expansion

Let $(X,\mathscr X)$ be a Borel or Polish space equipped with a $\sigma$-finite, non-atomic measure $\mu$. Let $\eta$ be a Poisson point process of intensity $\mu$, and $k \in \mathbb{N}$. Given a symmetric, measurable kernel $f: X^k \to \mathbb{R}$ with $f \in L^1_s(X^k, \mu^k)$, the Poisson U-statistic of order $k$ is

$(X,\mathscr X)$0

where $(X,\mathscr X)$1 denotes all ordered $(X,\mathscr X)$2-tuples of distinct points in the support of $(X,\mathscr X)$3 (Lachèze-Rey et al., 2015). For $(X,\mathscr X)$4 theory or chaos expansion, require $(X,\mathscr X)$5.

The crucial representation is the finite Wiener–Itô chaos expansion (Reitzner et al., 2011):

$(X,\mathscr X)$6

with projection kernels

$(X,\mathscr X)$7

for $(X,\mathscr X)$8; $(X,\mathscr X)$9 denotes the $\sigma$0 multiple Poisson integral with respect to the compensated process $\sigma$1. Higher-order contractions and cumulant formulas derive from this structure (Lachèze-Rey et al., 2015, Benes et al., 2014).

2. Poisson U-Statistics in Limit Theorems: Central Limit Theorems, Gaussian and Non-Gaussian Regimes

Poisson U-statistics admit precise central limit theorems in both univariate and multivariate settings. Suppose $\sigma$2 with $\sigma$3 a Poisson process of intensity $\sigma$4.

• As $\sigma$5, under mild integrability, normalization by $\sigma$6 yields asymptotic normality:

$\sigma$7

with analogous vector-valued limits for multivariate U-statistics and explicit covariance formulas involving contractions (Vecera et al., 2015). These limit theorems extend to local $\sigma$8-statistics in diverging domains (e.g., halfspaces), with rates quantified in the Kolmogorov or Wasserstein metrics (Thomas, 2022).

• The rate of convergence for geometric U-statistics (fixed $\sigma$9, intensity $\mu$0) is typically $\mu$1 in both Kolmogorov and Wasserstein distance, provided the first chaos dominates (Reitzner et al., 2011, Schulte, 2012). The rate improves for certain degenerate U-statistics or under additional regularity or "localization" of $\mu$2 (Bourguin et al., 2016, Bourguin et al., 2014).
• In the fully degenerate case (e.g., $\mu$3, $\mu$4 symmetric, $\mu$5), Gamma (as opposed to Gaussian) limiting behavior occurs under exact fourth-moment and contraction control (Peccati et al., 2013). Hybrid Gaussian-Gamma mixed limits (or Gamma-Poisson) are possible for multidimensional functionals composed of several U-statistics of differing degeneracy (Peccati et al., 2013).

3. Variance, Moment, and Cumulant Formulae

The variance of a Poisson U-statistic is given by

$\mu$6

where $\mu$7 are the Wiener–Itô projections of $\mu$8 (Lachèze-Rey et al., 2015, Reitzner et al., 2011). Higher moments and cumulants can be expressed as explicit combinatorial sums over partitions (diagram formula), generalizing the Mecke formula for Poisson processes and enabling sharp moment control (Benes et al., 2014, Vecera et al., 2015, Lachèze-Rey et al., 2015).

For $\mu$9, the variance can also be expanded explicitly as

$\eta$0

Contractions between kernels determine the leading terms in fourth moment and cumulant estimates, which are used to quantify normal or gamma approximation rates (Lachieze-Rey et al., 2012, Peccati et al., 2013).

4. Stein–Malliavin Approach: Quantitative Normal Approximations and Contraction Norms

The Stein–Malliavin method combines Poisson difference operators with Wiener–Itô analysis to bound distances to limiting distributions. For $\eta$1 centered, let $\eta$2 denote the add-one operator, $\eta$3 the Poisson Ornstein–Uhlenbeck generator, and $\eta$4 its pseudo-inverse (Minh, 2011, Bourguin et al., 2014).

Key bounds:

• Wasserstein distance:

$\eta$5

where $\eta$6 are explicit fourth moment functionals computed as sums of $\eta$7-norms of contractions of $\eta$8 (Reitzner et al., 2011, Schulte, 2012).

• Kolmogorov distance and $\eta$9 bounds can be similarly formulated; for multidimensional U-statistics, explicit covariance control via contractions is available (Minh, 2011). Rates are optimal up to constants for most geometric/statistical models.
• Fourth-moment and contraction conditions yield exact Berry–Esseen constants and allow for rates as precise as $\mu$0 in manifold- or wavelet-based U-statistics, where $\mu$1 is a smoothness parameter and $\mu$2 the ambient dimension (Bourguin et al., 2016, Bourguin et al., 2014).

5. Concentration, Large Deviations, and Optimal Tail Inequalities

Sharp concentration inequalities for Poisson U-statistics complement CLT behavior and control moderate to large deviations. General results state that, for $\mu$3 a Poisson U-statistic of order $\mu$4:

$\mu$5

with

$\mu$6

for large $\mu$7, where $\mu$8 is the process intensity (Bonnet et al., 2024). These bounds are proven optimal: no tail bound with exponent $\mu$9 is possible, reflecting a Poissonian rather than Gaussian large-deviation regime (Bachmann et al., 2015). The same order of bounds holds for functionals like subgraph counts in random geometric graphs and power-weighted edge-lengths, as well as for U-statistics associated to intrinsic volumes of intersection processes (Bonnet et al., 2024, Bachmann et al., 2015). Moderate deviations $k \in \mathbb{N}$0 yield Gaussian tails, again reflecting the transition from Poissonian to normal fluctuation regimes.

6. Functional Poisson Approximation and Laws of Iterated Logarithm

Beyond distributional approximations, Poisson U-statistics viewed as point processes can be functionally approximated by Poisson or compound Poisson processes in Kantorovich–Rubinstein distance (Decreusefond et al., 2014, Pianoforte et al., 2021). The error in such optimal-transport distances is computable from second-order remainder integrals, and the rates are shown to be $k \in \mathbb{N}$1 (for suitable scaling).

Moreover, exponential moment inequalities for Poisson U-statistics yield laws of the iterated logarithm (LIL). For a degenerate kernel of type $k \in \mathbb{N}$2, one has almost surely, as $k \in \mathbb{N}$3:

$k \in \mathbb{N}$4

with the precise limsup characterized via the chaos expansion (Adamczak et al., 2024). These asymptotics extend to classical examples such as subgraph counts, edge-length functionals, and Poisson-driven Ornstein–Uhlenbeck quadratic statistics.

7. Applications in Geometry, Manifolds, Statistical Inference, and Beyond

Applications of Poisson U-statistics include:

• Subgraph and simplex counts, power-weighted length functionals, and other geometric statistics in random geometric graphs on Euclidean, spherical, or manifold domains (Thomas, 2022, Bourguin et al., 2016, Bourguin et al., 2014).
• Estimation and denoising in nonparametric regression and density estimation over manifolds or the sphere, using needlet- and wavelet-based U-statistics (Bourguin et al., 2016, Bourguin et al., 2014).
• Intrinsic volumes, intersection counts, and other functionals of random hyperplane, facet, or plate processes, leading to explicit CLTs and rates depending on the kernel and dimension (Lachèze-Rey et al., 2015, Vecera et al., 2015).
• Explicit Poisson and compound Poisson approximations for derived point processes in geometric or combinatorial settings, with quantitative error control in transportation metrics (Decreusefond et al., 2014, Pianoforte et al., 2021).
• Multivariate, mixed, and hybrid limit theorems, with joint convergence to Gaussian, Poisson, and Gamma laws, depending on degeneracy and scaling (Peccati et al., 2013).

These results demonstrate that Poisson U-statistics form a central analytical and probabilistic tool, bridging combinatorics, spatial statistics, and stochastic process theory, with robust methodology for asymptotic analysis, concentration, and inference (Lachèze-Rey et al., 2015, Benes et al., 2014, Bourguin et al., 2016, Minh, 2011, Schulte, 2012, Vecera et al., 2015, Bonnet et al., 2024, Adamczak et al., 2024).