arXiv subjects

Raphaël Lachièze-Rey

Publications and source records attributed to Raphaël Lachièze-Rey.

27 records · Page 2Linked to original sources

arXiv2017

Bounds to the normal for proximity region graphs

In a proximity region graph ${\cal G}$ in $\mathbb{R}^d$, 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 disc 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 ${\cal G}$, 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.

math.PR↗

arXiv2017

Bicovariograms and Euler characteristic I. Regular sets

We establish an expression of the \EC~of a $r$-regular planar set in function of some variographic quantities. The usual $\mathcal{C} ^{2}$ framework is relaxed to a $\mathcal{C} ^{1,1}$ regularity assumption, generalising existing local formulas for the \EC. We give also general bounds on the number of connected components of a measurable set of $\mathbb{R}^{2}$ in terms of local quantities. These results are then combined to yield a new expression of the mean \EC~of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. We derive results for level sets of some moving average processes and for the boolean model with non-connected polyrectangular grains in $\mathbb{R}^{2}$. Applications to excursions of smooth bivariate random fields are derived in the companion paper \cite{LacEC2}, and applied for instance to $\C^{1,1}$ Gaussian fields, generalising standard results.

math.PR↗

arXiv2017

Normal approximation for stabilizing functionals

We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin-Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind. Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of $\mathbb{R}^d$, including $m$-dimensional Riemannian manifolds, $m\leq d$. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the $k$-face and $i$th intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension $d\geq 2$. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.

math.PR↗

arXiv2016

Two-dimensional Kac-Rice formula. Application to shot noise processes excursions

Given a deterministic function f:R^2->R atisfying suitable assump- tions, we show that for h smooth with compact support, the integral of the Euler characteristic of the excursion set of f above some level u against a test function h corresponds to the Lebesgue integral on R^2 of a bounded quantity depending on grad(f)(x),h(f(x)),h'(f(x)) and \partial_{ii}f(x),i = 1,2. This formula can be seen as a 2-dimensional analogue of Kac-Rice formula. It yields in particular that the left hand member is continuous in the argument f, for an appropriate norm on the space of C2 functions. If f is a random field, the expectation can be passed under integrals in this identity under minimal requirements, not involving any density assumptions on the marginals of f or his derivatives. We apply these results to give a weak expression of the mean Euler characteristic of a shot noise process, and the finiteness of its moments.

math.PR↗

arXiv2016

Limit theorems for a class of stationary increments Levy driven moving averages

In this paper we present some new limit theorems for power variation of $k$th order increments of stationary increments Lévy 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 very surprising results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we will show that the first order limit theorems and the mode of convergence strongly depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal--Getoor index $β\in (0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $α$. 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 the second order limit theorem connected to the ergodic type result. When the driving Lévy process $L$ is a symmetric $β$-stable process we obtain two different limits: a central limit theorem and convergence in distribution towards a $(1-α)β$-stable totally right skewed random variable.

math.PR↗

arXiv2015

Boundary density and Voronoi set estimation for irregular sets

In this paper, we study the inner and outer boundary densities of some sets with self-similar boundary having Minkowski dimension $s\textgreater{}d-1$ in $\mathbb{R}^{d}$. These quantities turn out to be crucial in some problems of set estimation theory, as we show here for the Voronoi approximation of the set with a random input constituted by $n$ iid points in some larger bounded domain. We prove that some classes of such sets have positive inner and outer boundary density, and therefore satisfy Berry-Essen bounds in $n^{-s/2d}$ for Kolmogorov distance. The Von Koch flake serves as an example, and a set with Cantor boundary as a counter-example. We also give the almost sure rate of convergence of Hausdorff distance between the set and its approximation.

math.PR↗

arXiv2015

Limit theorems for stationary increments Lévy driven moving averages

In this paper we present some new limit theorems for power variation of $k$th order increments of stationary increments Lévy driven moving averages. In this infill sampling setting, the asymptotic theory gives very surprising results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we will show that the first order limit theorems and the mode of convergence strongly depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal-Getoor index $β\in (0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $α$. First order asymptotic theory essentially comprise 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 the second order limit theorem connected to the ergodic type result. When the driving Lévy process $L$ is a symmetric $β$-stable process we obtain two different limits: a central limit theorem and convergence in distribution towards a $(1-α)β$-stable random variable.

math.PR↗

arXiv2015

New Kolmogorov bounds for functionals of binomial point processes

We obtain explicit Berry-Esseen bounds in the Kolmogorov distance 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 recently 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 covering processes.

math.PR↗

arXiv2009

Strong mixing property for STIT tessellations

The so-called STIT tessellations form the class of homogeneous (spatially stationary) tessellations of $\mathbb{R}^d$ which are stable under the nesting/iteration operation. In this paper, we establish the strong mixing property for these tessellations and give the optimal form of the rate of decay for the quantity $|\mathbb{P}({A}\cap Y=\emptyset,T_h B \cap Y=\emptyset)-\mathbb{P}({A}\cap Y=\emptyset)\mathbb{P}({B}\cap Y=\emptyset)|$ when $A$ and $B$ are two compact sets, $h$ a vector of $\mathbb{R}^d$, $T_{h}$ the corresponding translation operator and $Y$ a STIT Tessellation.

math.PR↗