Let (F n) be any sequence of Wiener chaoses of any fixed
order converging in distribution towards a standard Gaussian. In this talk, without any
additional assumptions, we shall explain how to derive the asymptotic smoothness of
the densities of F n , as well...
Nous présentons une nouvelle approche pour étudier la régularité de la loi d'une variable aléatoire quand l'espace de probabilité est équipé d'une forme de Dirichlet. Plus précisément nous développons une nouvelle technique pour contrôler les moments négatifs du carré du champ d'une variable aléatoire et utilisons le résultat (bien connu) qu'un tel contrôle implique un contrôle sur les normes...
Over the past decade, Entropic Optimal Transport problem has emerged as a versatile and computationally more tractable proxy for the Optimal Transport (Monge-Kantorovich) problem for applications in data science and statistical machine learning. One of the reasons behind the interest in adding an entropic penalty in the Monge Kantorovich problem is the fact that solutions can be computed by...
The Poincaré inequality governs the exponential convergence rate of algorithms such as Langevin dynamics. Interesting questions are then to understand how the Poincaré constant changes when the dynamics is perturbed, or to understand when this constant is minimal under certain constraints. In this talk, I will present some such results in the context of Markov diffusions. Their proof is based...
Joint work with N. Privault and A. Réveillac
We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals of deterministic and non-negative integrands with respect to Hawkes processes by a normally distributed random variable. Our results are specifically applied to compound Hawkes processes, and improve on the current literature where estimates...
Let $K$ be a convex body in $\mathbb{R}^d$. Let $X_K$ be a $d$-dimensional random vector distributed according to the Hadwiger-Wills density $\mu_K$ associated with $K$, defined as $\mu_K(x)=ce^{-\pi {\rm dist}^2(x,K)}$, $x\in \mathbb{R}^d$. Finally, let the information content $H_K$ be defined as $H_K={\rm dist}^2(X_K,K)$.
In this talk, we will study the fluctuations of $H_K$ around its...
Joint work with L. Coutin
Solving the SDE $dX(t)=r(X(t)) dt + dB(t) (1)$ is equivalent invert the map $B\mapsto B(t)-\int_0^t r(B(s)) ds$.
We study the analog of this problem on the Poisson space. Because of the Girsanov Theorem, it turns out that equivalent problem consists in inverting a time change.
We can then reinterpret the solution of the generalized Hawkes problem (find a self...
Stein's method applied to orthogonal decompositions has allowed to establish second order Poincaré inequalities for random functionals of binomial input and Poisson input. We will show how to apply these inequalities, and in particular how they enabled to show limit theorems for geometric functionals for random processes defined on the Euclidean space or a smooth manifold.
In this talk, I will present new stability results for non-degenerate centered self-decomposable laws with finite second moment and for non-degenerate symmetric alpha-stable laws with alpha in (1,2). These stability results are based on Stein's method and closed forms techniques.
As an application, explicit rates of convergence are obtained for several instances of the generalized CLTs.
We present some quantitative bounds on the so-called Stein magic factors of discrete distributions. These ones are obtained from intertwining relations between Markov semigroups of birth-death processes and discrete gradients. We also illustrate the application of this Stein magic factors for the convergence of the binomial negative law to the Poisson one.
We introduce a model of Poisson random waves in S^2 and we study Quantitative Central Limit Theorems when both the rate of the Poisson process and the frequency of the waves (eigenfunctions) diverge to infinity. We consider finite-dimensional distributions, harmonic coefficients and convergence in law in functional spaces, and we investigate carefully the interplay between the rate of...
