Raphaël Lachièze-Rey

Regularity conditions in the realisability problem with applications to point processes and random closed sets

We study existence of random elements with partially specified distributions. The technique relies on the existence of a positive ex-tension for linear functionals accompanied by additional conditions that ensure the regularity of the extension needed for interpreting it as a probability measure. It is shown in which case the extens ion can be chosen to possess some invariance properties. The results are applied to the existence of point processes with given correlation measure and random closed sets with given two-point covering function or contact distribution function. It is shown that the regularity condition can be efficiently checked in many cases in order to ensure that the obtained point processes are indeed locally finite and random sets have closed realisations.

Power variation for a class of stationary increments Lévy driven moving averages

In this paper, we present some new limit theorems for power variation of kkth order increments of stationary increments Levy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k≥1k≥1, the considered power p>0p>0, the Blumenthal–Getoor index β∈[0,2)β∈[0,2) of the driving pure jump Levy process LL and the behaviour of the kernel function gg at 00 determined by the power αα. First-order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove a second-order limit theorem connected to the ergodic type result. When the driving Levy process LL is a symmetric ββ-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k−α)β(k−α)β-stable totally right skewed random variable.

U-statistics in stochastic geometry

A U-statistic of order k with kernel \(f: \mathbb{X}^{k} \rightarrow \mathbb{R}^{d}\) over a Poisson process η is defined as

New Berry-Esseen bounds for functionals of binomial point processes

\displaystyle{\sum _{(x_{1},\ldots,x_{k})}f(x_{1},\ldots,x_{k}),}

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

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, characteristics of random geometric graphs, volumes of random simplices, and many others. It turns out that the Wiener–Ito chaos expansion of a U-statistic is finite and thus Malliavin calculus is a particularly suitable method. Variance estimates, approximation of the covariance structure, and limit theorems which have been out of reach for many years can be derived. In this chapter we state the fundamental properties of U-statistics and investigate moment formulae. The main object of the chapter is to introduce the available limit theorems.

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

This paper concerns the second-order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, along with an algorithmic implementation, and illustrate our study with two theoretical applications: 1 the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and 2 not every covariance/indicator variogram can be obtained with a Gaussian level set.

New Kolmogorov bounds for functionals of binomial point processes

We obtain explicit Berry-Esseen bounds in the Kolmogorov dis- tance for the normal approximation of non-linear functionals of vectors of independent random variables. Our results are based on the use of Stein’s method and of random difference operators, and generalise the bounds obtained by Chatterjee (2008), concerning normal approximations in the Wasserstein distance. In order to obtain lower bounds for variances, we also revisit the classical Hoeffding decompositions, for which we provide a new proof and a new representation. Several applications are discussed in detail: in particular, new Berry-Esseen bounds are obtained for set approximations with random tessellations, as well as for functionals of coverage processes.

Rearrangements of Gaussian fields

In a proximity region graph G in Rd, two distinct points x,y of a point process μ are connected when the ‘forbidden region’ S(x,y) these points determine has empty intersection with μ. The Gabriel graph, where S(x,y) is the open disk with diameter the line segment connecting x and y, is one canonical example. When μ is a Poisson or binomial process, under broad conditions on the regions S(x,y), bounds on the Kolmogorov and Wasserstein distances to the normal are produced for functionals of G, including the total number of edges and the total length. Variance lower bounds, not requiring strong stabilization, are also proven to hold for a class of such functionals.