arXiv subjects

Céline Duval

Publications and source records attributed to Céline Duval.

14 recordsLinked to original sources

arXiv2026

Local stability and rates of convergence to equilibrium for the Nonlinear Renewal Equation; applications to Hawkes processes

We study the asymptotic properties of the solutions of a nonlinear renewal equation. The main contribution of the present article is to provide stability and convergence results around equilibrium solutions, under some local subcritical condition. Quantitative rates of convergence to equilibrium are established. Instability results are given in both the critical and supercritical cases. As an implication of these results, we establish a Central Limit Theorem for Hawkes processes in a mean-field interaction.

math.DS↗

arXiv2024

Nonparametric density estimation for the small jumps of Lévy processes

We consider the problem of estimating the density of the process associated with the small jumps of a pure jump Lévy process, possibly of infinite variation, from discrete observations of one trajectory. The interest of such a question lies on the observation that even when the Lévy measure is known, the density of the increments of the small jumps of the process cannot be computed in closed-form. We discuss results both from low and high frequency observations. In a low frequency setting, assuming the Lévy density associated with the jumps larger than $\varepsilon\in (0,1]$ in absolute value is known, a spectral estimator relying on the convolution structure of the problem achieves a parametric rate of convergence with respect to the integrated $L_2$ loss, up to a logarithmic factor. In a high frequency setting, we remove the assumption on the knowledge of the Lévy measure of the large jumps and show that the rate of convergence depends both on the sampling scheme and on the behaviour of the Lévy measure in a neighborhood of zero. We show that the rate we find is minimax up to a logarithmic factor. An adaptive penalized procedure is studied to select the cutoff parameter. These results are extended to encompass the case where a Brownian component is present in the Lévy process. Furthermore, we illustrate numerically the performances of our procedures.

math.ST↗

arXiv2024

Adaptive minimax estimation for discretely observed Lévy processes

In this paper, we study the nonparametric estimation of the density $f_Δ$ of an increment of a Lévy process $X$ based on $n$ observations with a sampling rate $Δ$. The class of Lévy processes considered is broad, including both processes with a Gaussian component and pure jump processes. A key focus is on processes where $f_Δ$ is smooth for all $Δ$. We introduce a spectral estimator of $f_Δ$ and derive both upper and lower bounds, showing that the estimator is minimax optimal in both low- and high-frequency regimes. Our results differ from existing work by offering weaker, easily verifiable assumptions and providing non-asymptotic results that explicitly depend on $Δ$. In low-frequency settings, we recover parametric convergence rates, while in high-frequency settings, we identify two regimes based on whether the Gaussian or jump components dominate. The rates of convergence are closely tied to the jump activity, with continuity between the Gaussian case and more general jump processes. Additionally, we propose a fully data-driven estimator with proven simplicity and rapid implementation, supported by numerical experiments.

math.ST↗

arXiv2024

Adaptation to inhomogeneous smoothness for densities with irregularities

We estimate on a compact interval densities with isolated irregularities, such as discontinuities or discontinuities in some derivatives. From independent and identically distributed observations we construct a kernel estimator with non-constant bandwidth, in particular in the vicinity of irregularities. It attains faster rates, for the risk $L_{p}, p\geq 1$, than usual estimators with a fixed global bandwidth. Optimality of the rate found is established by a lower bound result. We then propose an adaptive method inspired by Lepski's method for automatically selecting the variable bandwidth, without any knowledge of the regularity of the density nor of the points where the regularity breaks down. The procedure is illustrated numerically on examples.

math.ST↗

arXiv2023

Adaptive directional estimator of the density in R^d for independent and mixing sequences

A new multivariate density estimator for stationary sequences is obtained by Fourier inversion of the thresholded empirical characteristic function. This estimator does not depend on the choice of parameters related to the smoothness of the density; it is directly adaptive. We establish oracle inequalities valid for independent, $α$-mixing and $τ$-mixing sequences, which allows us to derive optimal convergence rates, up to a logarithmic loss. On general anisotropic Sobolev classes, the estimator adapts to the regularity of the unknown density but also achieves directional adaptivity. In particular, if A is an invertible matrix, if the observations are drawn from X $\in$ R^d , d $\ge$ 1, it achieves the rate implied by the regularity of AX, which may be more regular than X. The estimator is easy to implement and numerically efficient. It depends on the calibration of a parameter for which we propose an innovative numerical selection procedure, using the Euler characteristic of the thresholded areas.

math.ST↗

arXiv2022

Surface area and volume of excursion sets observed on point cloud based polytopic tessellations

The excursion set of a $C^2$ smooth random field carries relevant information in its various geometric measures. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but rather to observations at discrete points in space. It has been reported that for specific regular lattices of points in dimensions 2 and 3, the usual estimate of the surface area of the excursions remains biased even when the lattice becomes dense in the domain of observation. In the present work, under the key assumptions of stationarity and isotropy, we demonstrate that this limiting bias is invariant to the locations of the observation points. Indeed, we identify an explicit formula for the bias, showing that it only depends on the spatial dimension $d$. This enables us to define an unbiased estimator for the surface area of excursion sets that are approximated by general tessellations of polytopes in $\mathbb{R}^d$, including Poisson-Voronoi tessellations. We also establish a joint central limit theorem for the surface area and volume estimates of excursion sets observed over hypercubic lattices.

math.PR↗

arXiv2021

Interacting Hawkes processes with multiplicative inhibition

In the present work, we introduce a general class of mean-field interacting nonlinear Hawkes processes modelling the reciprocal interactions between two neuronal populations, one excitatory and one inhibitory. The model incorporates two features: inhibition, which acts as a multiplicative factor onto the intensity of the excitatory population and additive retroaction from the excitatory neurons onto the inhibitory ones. We give first a detailed analysis of the well-posedness of this interacting system as well as its dynamics in large population. The second aim of the paper is to give a rigorous analysis of the longtime behavior of the mean-field limit process. We provide also numerical evidence that inhibition and retroaction may be responsible for the emergence of limit cycles in such system.

math.PR↗

arXiv2020

Spectral-free estimation of Lévy densities in high-frequency regime

We construct an estimator of the Lévy density of a pure jump Lévy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the Lévy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the Lévy--Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of $L_p$ loss functions, $p\geq 1$, over Besov balls. We recover classical nonparametric rates for finite variation Lévy processes and for a large nonparametric class of symmetric infinite variation Lévy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.

math.PR↗

arXiv2020

Non-asymptotic control of the cumulative distribution function of Lévy processes

We propose non-asymptotic controls of the cumulative distribution function $P(|X_{t}|\ge \varepsilon)$, for any $t>0$, $\varepsilon>0$ and any Lévy process $X$ such that its Lévy density is bounded from above by the density of an $α$-stable type Lévy process in a neighborhood of the origin. The results presented are non-asymptotic and optimal, they apply to a large class of Lévy processes.

math.PR↗

arXiv2019

Total variation distance for discretely observed Lévy processes: a Gaussian approximation of the small jumps

It is common practice to treat small jumps of Lévy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistically speaking. We elaborate a fine analysis of a Gaussian approximation for the small jumps of Lévy processes with infinite Lévy measure in total variation distance. Non asymptotic bounds for the total variation distance between $n$ discrete observations of small jumps of a Lévy process and the corresponding Gaussian distribution is presented and extensively discussed. As a byproduct, new upper bounds for the total variation distance between discrete observations of Lévy processes are provided. The theory is illustrated by concrete examples.

math.ST↗

arXiv2018

An adaptive procedure for Fourier estimators: illustration to deconvolution and decompounding

We introduce a new procedure to select the optimal cutoff parameter for Fourier density estimators that leads to adaptive rate optimal estimators, up to a logarithmic factor. This adaptive procedure applies for different inverse problems. We illustrate it on two classical examples: deconvolution and decompounding, i.e. non-parametric estimation of the jump density of a compound Poisson process from the observation of n increments of length $Δ$ > 0. For this latter example, we first build an estimator for which we provide an upper bound for its L 2-risk that is valid simultaneously for sampling rates $Δ$ that can vanish, $Δ$ := $Δ$ n $\rightarrow$ 0, can be fixed, $Δ$ n $\rightarrow$ $Δ$ 0 > 0 or can get large, $Δ$ n $\rightarrow$ $\infty$ slowly. This last result is new and presents interest on its own. Then, we show that the adaptive procedure we present leads to an adaptive and rate optimal (up to a logarithmic factor) estimator of the jump density.

math.ST↗

arXiv2016

Nonparametric adaptive estimation for grouped data

The aim of this paper is to estimate the density f of a random variable X when one has access to independent observations of the sum of K $\ge$ 2 independent copies of X. We provide a constructive estimator based on a suitable definition of the logarithm of the empirical characteristic function.We propose a new strategy for the data driven choice of the cut-off parameter. The adaptive estimator is proven to be minimax-optimal up to some logarithmic loss. A numerical study illustrates the performances of the method. Moreover, we discuss the fact that the definition of the estimator applies in a wider context than the one considered here.

math.ST↗

arXiv2012

Adaptive wavelet estimation of a compound Poisson process

We study the nonparametric estimation of the jump density of a compound Poisson process from the discrete observation of one trajectory over $[0,T]$. We consider the microscopic regime when the sampling rate $Δ=Δ_T\rightarrow0$ as $T\rightarrow\infty$. We propose an adaptive wavelet threshold density estimator and study its performance for the $L_p$ loss, $p\geq 1$, over Besov spaces. The main novelty is that we achieve minimax rates of convergence for sampling rates $Δ_T$ that vanish with $T$ at arbitrary polynomial rates. More precicely, our estimator attains minimax rates of convergence provided there exists a constant $K\geq 1$ such that the sampling rate $Δ_T$ satisfies $TΔ_T^{2K+2}\leq 1.$ If this condition cannot be satisfied we still provide an upper bound for our estimator. The estimating procedure is based on the inversion of the compounding operator in the same spirit as Buchmann and Grübel (2003).

math.ST↗

arXiv2011

Statistical inference across time scales

We investigate statistical inference across time scales. We take as toy model the estimation of the intensity of a discretely observed compound Poisson process with symmetric Bernoulli jumps. We have data at different time scales: microscopic, intermediate and macroscopic. We quantify the smooth statistical transition from a microscopic Poissonian regime to a macroscopic Gaussian regime. The classical quadratic variation estimator is efficient in both microscopic and macroscopic scales but surprisingly shows a substantial loss of information in the intermediate scale that can be explicitly related to the sampling rate. We discuss the implications of these findings beyond this idealised framework.

math.ST↗