arXiv subjects

Delphine Féral

Publications and source records attributed to Delphine Féral.

8 recordsLinked to original sources

arXiv2026

Large time behaviour for the semigroup of the kinetic Brownian motion in the plane

We establish an integration by parts formula for the semigroup in time $T>0$ of the kinetic Brownian motion in the Euclidean plane together with its velocity in the circle. The stochastic differential equation of our kinetic Brownian motion is driven here by one real-valued Brownian motion constructed from an orthonormal basis of $L^2([0,T],\R)$ and an independent sequence of $\SN(0,1)$ random variables. Our method is based on an explicit computation of a Malliavin dual in the Gaussian space. We are mainly interested in large time $T$. From our integration by parts, we obtain gradient estimates including a reverse Poincar{é} inequality for the semigroup. As a direct consequence, we also obtain a Liouville property for the generator of the kinetic Brownian motion and its speed: all bounded harmonic functions are constant.

math.PR↗

arXiv2024

A coupling strategy for Brownian motions at fixed time on Carnot groups using Legendre expansion

We propose a new simple construction of a coupling at a fixed time of two sub-Riemannian Brownian motions on the Heisenberg group and on the free step 2 Carnot groups. The construction is based on a Legendre expansion of the standard Brownian motion and of the L{é}vy area. We deduce sharp estimates for the decay in total variation distance between the laws of the Brownian motions. Using a change of probability method, we also obtain the log-Harnack inequality, a Bismut type integration by part formula and reverse Poincaré inequalities for the associated semi-group.

math.PR↗

arXiv2017

Generalized SURE for optimal shrinkage of singular values in low-rank matrix denoising

We consider the problem of estimating a low-rank signal matrix from noisy measurements under the assumption that the distribution of the data matrix belongs to an exponential family. In this setting, we derive generalized Stein's unbiased risk estimation (SURE) formulas that hold for any spectral estimators which shrink or threshold the singular values of the data matrix. This leads to new data-driven spectral estimators, whose optimality is discussed using tools from random matrix theory and through numerical experiments. Under the spiked population model and in the asymptotic setting where the dimensions of the data matrix are let going to infinity, some theoretical properties of our approach are compared to recent results on asymptotically optimal shrinking rules for Gaussian noise. It also leads to new procedures for singular values shrinkage in finite-dimensional matrix denoising for Gamma-distributed and Poisson-distributed measurements.

math.ST↗

arXiv2011

Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices

We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner matrices when the dimension goes to infinity. The entries of the Hermitian Wigner matrix have a distribution which is symmetric and satisfies a Poincaré inequality. The perturbation matrix is a deterministic Hermitian matrix whose spectral measure converges to some probability measure with compact support. We assume that this perturbation matrix has a fixed number of fixed eigenvalues (spikes) outside the support of its limiting spectral measure whereas the distance between the other eigenvalues and this support uniformly goes to zero as the dimension goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of the deformed model which will converge to some limiting points outside the support of the limiting spectral measure. This phenomenon can be fully described in terms of free probability involving the subordination function related to the additive free convolution of the limiting spectral measure of the perturbation matrix by a semi-circular distribution. Note that up to now only finite rank perturbations had been considered (even in the deformed GUE case).

math.PR↗

arXiv2011

Central limit theorems for eigenvalues of deformations of Wigner matrices

In this paper, we explain the dependance of the fluctuations of the largest eigenvalues of a Deformed Wigner model with respect to the eigenvectors of the perturbation matrix. We exhibit quite general situations that will give rise to universality or non universality of the fluctuations.

math.PR↗

arXiv2011

The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations

In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices $(M_N)_N$ defined by $M_N=W_N/\sqrt{N}+A_N$ where $W_N$ is an $N\times N$ Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix $A_N$ is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of $A_N$ are sufficiently far from zero, the corresponding eigenvalues of $M_N$ almost surely exit the limiting semicircle compact support as the size $N$ becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of $W_N$. On the other hand, when $A_N$ is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of $W_N$.

math.PR↗

arXiv2008

The largest eigenvalues of sample covariance matrices for a spiked population: diagonal case

We consider large complex random sample covariance matrices obtained from "spiked populations", that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to one. We investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large. Under some conditions on moments of the sample distribution, we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex Gaussian sample with the same true covariance. The real setting is also considered.

math-ph↗

arXiv2006

The largest eigenvalue of rank one deformation of large Wigner matrices

The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.

math.PR↗