arXiv subjects

Nicolas Chenavier

Publications and source records attributed to Nicolas Chenavier.

At least 19 recordsLinked to original sources

arXiv2025

Voronoi integration of the rendering equation

In photorealistic image rendering, Monte Carlo methods form the foundation for the integration of the rendering equation in modern approaches. However, despite their effectiveness, traditional Monte Carlo methods often face challenges in controlling variance, resulting in noisy visual artifacts in regions that are difficult to render. In this work, we propose a new approach to the integration of the rendering equation by introducing a Voronoi tessellation reweighting scheme combined with a Poisson point process sampling strategy to address some of the limitations of standard Monte Carlo methods. From a theoretical point of view, we show that the variance induced by a Poisson-Voronoi tessellation is smaller than that of the Monte Carlo method when the intensity of the underlying process is arbitrarily large and when the function to be integrated satisfies a Holder continuity condition.

math.NA↗

arXiv2025

Construction of ergodic IDLA forests in $\mathbb{Z}^d$

We prove the existence of infinite-volume IDLA forests in $\mathbb{Z}^d$ , with $d \geq 2$, based on a multi-source IDLA protocol. Unlike IDLA aggregates, the laws of the IDLA forests studied here depend on the trajectories of particles, and then do not satisfy the famous Abelian property. Their existence is due to a stabilization result (Theorem 1.1, our main result) that we establish using percolation tools. Although the sources are infinitely many, we also prove that each of them play the same role in the building procedure, which results in an ergodicity property for the IDLA forests (Theorem 1.2).

math.PR↗

arXiv2025

Central limit theorems for squared increment sums of fractional Brownian fields based on a Delaunay triangulation in $2D$

An isotropic fractional Brownian field (with Hurst parameter $H<1/2$) is observed in a family of points in the unit square $\mathbf{C}=(-1/2,1/2]^{2}$% . These points are assumed to come from a realization of a homogeneous Poisson point process with intensity $N$. We consider normalized increments (resp. pairs of increments) along the edges of the Delaunay triangulation generated by the Poisson point process (resp. pairs of edges within triangles). Central limit theorems are established for the respective centered squared increment sums as $N\rightarrow \infty $.

math.PR↗

arXiv2025

Limit theorems for squared increment sums of the maximum of two isotropic fractional Brownian fields under fixed-domain asymptotics

We study squared increment sums of the pointwise maximum of two independent and identically distributed isotropic fractional Brownian fields over a fixed two-dimensional domain. The fields are observed at the points of a homogeneous Poisson point process with intensity $N$, and increments are computed along the edges of the associated Delaunay triangulation. In contrast with the case of a single fractional Brownian field, where centered squared increment sums satisfy a central limit theorem after the usual normalization, the pointwise maximum exhibits a different asymptotic regime. The dominant contribution comes from Delaunay edges located in a shrinking neighborhood of the random interface where the two fractional Brownian fields exchange the role of the maximizer. For Hurst parameter $H<1/2$, we prove that the properly normalized squared increment sum converges in probability to a deterministic constant times the local time at zero of the difference between the two fields. This shows that the asymptotic behavior is governed by the geometry of the random contact set rather than by Gaussian fluctuation effects. The result provides a key ingredient for fixed-domain asymptotic inference in Brown--Resnick type models based on randomly located observations.

math.PR↗

arXiv2025

Asymptotic properties of maximum composite likelihood estimators for max-stable Brown-Resnick random fields over a fixed-domain

Likelihood-based inference for max-stable random fields is challenging, since finite-dimensional densities are either unavailable in closed form or computationally intractable in moderate to high dimension. Composite likelihood methods, based on low-dimensional marginal densities, therefore provide a natural alternative. In this paper, we study maximum composite likelihood estimation for spatial Brown--Resnick random fields generated by isotropic fractional Brownian fields. We work under fixed-domain asymptotics: a single realization of the max-stable field is observed on an increasingly dense random set of sites, given by a homogeneous Poisson point process. Pairwise and triplewise composite likelihoods are constructed by retaining, respectively, the edges and the triangles of the associated Poisson--Delaunay triangulation. Our main results establish the consistency of the resulting maximum composite likelihood estimators of the scale and smoothness parameters, when the other parameter is known. Their asymptotic behaviour is non-standard: the estimators converge at rates depending on the smoothness parameter and their centered limits are non-Gaussian. More precisely, the limiting fluctuations are driven by aggregated local times associated with the canonical tessellation of the Brown--Resnick field. These results reveal a fundamental departure from the classical composite likelihood theory based on increasing domains or independent replications, and show that Gaussian uncertainty quantification may be misleading in fixed-domain inference for max-stable spatial extremes.

math.ST↗

arXiv2024

IDLA with sources in a hyperplane of $\mathbb{Z}^d$

We consider a random growth model based on the IDLA protocol with sources in a hyperplane of $Z^d$ . We provide a stabilization result and a shape theorem generalizing [7] in any dimension by introducing new techniques leading to a rough global upper bound.

math.PR↗

arXiv2023

Compound Poisson process approximation under $\beta$-mixing and stabilization

We establish Poisson and compound Poisson approximations for stabilizing statistics of $\beta$-mixing point processes and give explicit rates of convergence. Our findings are based on a general estimate of the total variation distance of a stationary $\beta$-mixing process and its Palm version. As main contributions, this article (i) extends recent results on Poisson process approximation to non-Poisson/binomial input, (ii) gives concrete bounds for compound Poisson process approximation in a Wasserstein distance and (iii) illustrates the applicability of the general result in an example on minimal angles in the stationary Poisson-Delaunay tessellation. The latter is among the first (nontrivial) situations in Stochastic Geometry, where compound Poisson approximation can be established with explicit extremal index and cluster size distribution.

math.PR↗

arXiv2022

Compound Poisson approximation for simple transient random walks in random sceneries

Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(\xi(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(\xi(S_n))_{n\geq 0}$. Under suitable conditions, we make explicit the extremal index and show that the point process of exceedances converges to a compound Poisson point process. We give two examples for which the cluster size distribution can be made explicit.

math.PR↗

arXiv2022

Some properties on extremes for transient random walks in random sceneries

Let $(S_n)_{n \geq 0}$ be a transient random walk in the domain of attraction of a stable law and let $(\xi(s))_{s \in \mathbb{Z}}$ be a stationary sequence of random variables. In a previous work, under conditions of type $D(u_n)$ and $D'(u_n)$, we established a limit theorem for the maximum of the first $n$ terms of the sequence $(\xi(S_n))_{n\geq 0}$ as $n$ goes to infinity. In this paper we show that, under the same conditions and under a suitable scaling, the point process of exceedances converges to a Poisson point process. We also give some properties of $(\xi(S_n))_{n\geq 0}$.

math.PR↗

arXiv2022

Fixed-domain asymptotic properties of maximum composite likelihood estimators for max-stable Brown-Resnick random fields

Likelihood inference for max-stable random fields is in general impossible because their finite-dimen\-sional probability density functions are unknown or cannot be computed efficiently. The weighted composite likelihood approach that utilizes lower dimensional marginal likelihoods (typically pairs or triples of sites that are not too distant) is rather favored. In this paper, we consider the family of spatial max-stable Brown-Resnick random fields associated with isotropic fractional Brownian fields. We assume that the sites are given by only one realization of a homogeneous Poisson point process restricted to $\mathbf{C}=(-1/2,1/2]^{2}$ and that the random field is observed at these sites. As the intensity increases, we study the asymptotic properties of the composite likelihood estimators of the scale and Hurst parameters of the fractional Brownian fields using different weighting strategies: we exclude either pairs that are not edges of the Delaunay triangulation or triples that are not vertices of triangles.

math.ST↗

arXiv2021

Limit laws for large kth-nearest neighbor balls

Let $X_1,\ldots,X_n$ be a sequence of independent random points in $\mathbb{R}^d$ with common Lebesgue density $f$. Under some conditions on $f$, we obtain a Poisson limit theorem, as $n \to \infty$, for the number of large probability $k$th-nearest neighbor balls of $X_1,\ldots,X_n$. Our result generalizes Theorem 2. of [10], which refers to the special case $k=1$. Our proof is completely different since it employs the Chen-Stein method instead of the method of moments. Moreover, we obtain a rate of convergence for the Poisson approximation.

math.PR↗

arXiv2020

Extremal life times of persistent loops and holes

Persistent homology captures the appearances and disappearances of topological features such as loops and holes when growing disks centered at a Poisson point process. We study extreme values for the life times of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation. First, we describe the scaling of the minimal life times for general feature dimensions, and of the maximal life times for holes in the Čech complex. Then, we proceed to a more refined analysis and establish Poisson approximation for large life times of holes and for small life times of loops. Finally, we also study the scaling of minimal life times in the Vietoris-Rips setting and point to a surprising difference to the Čech complex.

math.PR↗

arXiv2020

The bi-dimensional Directed IDLA forest

We investigate three types of Internal Diffusion Limited Aggregation (IDLA) models. These models are based on simple random walks on $\mathbf{Z}^2$ with infinitely many sources that are the points of the vertical axis $I(\infty)=\{0\}\times\mathbf{Z}$. Various properties are provided, such as stationarity, mixing, stabilization and shape theorems. Our results allow us to define a new directed (w.r.t. the horizontal direction) random forest spanning $\mathbf{Z}^2$, based on an IDLA protocol, which is invariant in distribution w.r.t. vertical translations.

math.PR↗

arXiv2019

Extremes for transient random walks in random sceneries under weak independence conditions

Let $\{ξ(k), k \in \mathbb{Z} \}$ be a stationary sequence of random variables with conditions of type $D(u_n)$ and $D'(u_n)$. Let $\{S_n, n \in \mathbb{N} \}$ be a transient random walk in the domain of attraction of a stable law. We provide a limit theorem for the maximum of the first $n$ terms of the sequence $\{ξ(S_n), n \in \mathbb{N} \}$ as $n$ goes to infinity. This paper extends a result due to Franke and Saigo who dealt with the case where the sequence $\{ξ(k), k \in \mathbb{Z} \}$ is i.i.d.

math.PR↗

arXiv2019

Testing goodness of fit for point processes via topological data analysis

We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions for a functional central limit theorem on bounded persistent Betti numbers of point processes with exponential decay of correlations.

math.ST↗

arXiv2018

The largest order statistics for the inradius in an isotropic STIT tessellation

A planar stationary and isotropic STIT tessellation at time $t>0$ is observed in the window $W_ρ={t^{-1}}\sqrt{π\ ρ}\cdot [-\frac{1}{2},\frac{1}{2}]^2$, for $ρ>0$. With each cell of the tessellation, we associate the inradius, which is the radius of the largest disk contained in the cell. Using the Chen-Stein method, we compute the limit distributions of the largest order statistics for the inradii of all cells whose nuclei are contained in $W_ρ$ as $ρ$ goes to infinity.

math.PR↗

arXiv2018

The maximal degree in a Poisson-Delaunay graph

We investigate the maximal degree in a Poisson-Delaunay graph in $\mathbf{R}^d$, $d\geq 2$, over all nodes in the window $\mathbf{W}_ρ:= ρ^{1/d}[0,1]^d$ as $ρ$ goes to infinity. The exact order of this maximum is provided in any dimension. In the particular setting $d=2$, we show that this quantity is concentrated on two consecutive integers with high probability. An extension of this result is discussed when $d\geq 3$

math.PR↗

arXiv2017

On the discrepancy of powers of random variables

Let $(d_n)$ be a sequence of positive numbers and let $(X_n)$ be a sequence of positive independent random variables. We provide an upper bound for the deviation between the distribution of the mantissaes of $(X_n^{d_n})$ and the Benford's law. If $d_n$ goes to infinity at a rate at most polynomial, this deviation converges a.s. to 0 as $N$ goes to infinity.

math.PR↗