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… 

Generalized limit theorems for U-max statistics

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

Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

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

The multivariate functional de Jong CLT

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

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

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

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

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

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

Introduction to Stochastic Geometry

This chapter introduces some of the fundamental notions from stochastic geometry. Background information from convex geometry is provided as far as this is required for the applications to stochastic

The Malliavin–Stein Method on the Poisson Space

This chapter provides a detailed and unified discussion of a collection of recently introduced techniques, allowing one to establish limit theorems with explicit rates of convergence, by combining

References

SHOWING 1-10 OF 37 REFERENCES

Central limit theorems for $U$-statistics of Poisson point processes

Central limit theorems for $U$-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable and the length of a random geometric graph are investigated.

Central limit theorems for Poisson hyperplane tessellations

We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in $\mathbb{R}^d$. This result generalizes an earlier one proved by

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

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

New Berry-Esseen bounds for non-linear functionals of Poisson random measures

This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein's method

Gamma limits and U-statistics on the Poisson space

Using Stein's method and the Malliavin calculus of variations, we derive explicit estimates for the Gamma approximation of functionals of a Poisson measure. In particular, conditions are presented

Normal Approximation of Poisson Functionals in Kolmogorov Distance

This paper shows that convergence in the Wasserstein distance of a Poisson functional and a Gaussian random variable has the same rate for both distances for a large class of Poisson functionals, namely so-called U-statistics ofPoisson point processes.

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with