# U-Statistics in Stochastic Geometry

@article{LachezeRey2015UStatisticsIS, title={U-Statistics in Stochastic Geometry}, author={Raphael Lacheze-Rey and Matthias Reitzner}, journal={arXiv: Probability}, year={2015}, volume={7}, pages={229-253} }

A U-statistic of order k with kernel \(f: \mathbb{X}^{k} \rightarrow \mathbb{R}^{d}\) over a Poisson process η is defined as
$$\displaystyle{\sum _{(x_{1},\ldots,x_{k})}f(x_{1},\ldots,x_{k}),}$$
where the summation is over k-tuples of distinct points of η, under appropriate integrability assumptions on f. U-statistics play an important role in stochastic geometry since many interesting functionals can be written as U-statistics, like intrinsic volumes of intersection processes…

## 14 Citations

### Generalized limit theorems for U-max statistics

- MathematicsJournal of Applied Probability
- 2022

Abstract
$U{\hbox{-}}\textrm{max}$
statistics were introduced by Lao and Mayer in 2008. Such statistics are natural in stochastic geometry. Examples are the maximal perimeters and areas of polygons…

### Concentration for Poisson U-statistics: Subgraph counts in random geometric graphs

- MathematicsStochastic Processes and their Applications
- 2018

### Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

- Mathematics
- 2015

Let η t be a Poisson point process with intensity measure tμ, t > 0, over a Borel space \(\mathbb{X}\), where μ is a fixed measure. Another point process ξ t on the real line is constructed by…

### Limit theory for U-statistics under geometric and topological constraints with rare events

- MathematicsJournal of Applied Probability
- 2022

Abstract We study the geometric and topological features of U-statistics of order k when the k-tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate…

### The multivariate functional de Jong CLT

- MathematicsProbability Theory and Related Fields
- 2022

We prove a multivariate functional version of de Jong’s CLT (J Multivar Anal 34(2):275–289, 1990) yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding…

### Large deviation principle for geometric and topological functionals and associated point processes

- Mathematics
- 2022

We prove a large deviation principle for the point process associated to k -element connected components in R d with respect to the connectivity radii r n → ∞ . The random points are generated from a…

### LARGE DEVIATION PRINCIPLE FOR GEOMETRIC AND TOPOLOGICAL FUNCTIONALS AND ASSOCIATED POINT PROCESSES

- Mathematics
- 2022

We prove a large deviation principle for the point process associated to k -element connected components in R d with respect to the connectivity radii r n → ∞ . The random points are generated from a…

### The fourth moment theorem on the Poisson space

- MathematicsThe Annals of Probability
- 2018

We prove an exact fourth moment bound for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result -- that has been elusive for…

### Concentration bounds for geometric Poisson functionals: Logarithmic Sobolev inequalities revisited

- Mathematics
- 2015

We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications,…

### U-Statistics on the Spherical Poisson Space

- Mathematics
- 2016

We review a recent stream of research on normal approximations for linear functionals and more general U-statistics of wavelets/needlets coefficients evaluated on a homogeneous spherical Poisson…

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