Workshop MESA - Stein's Method and Applications

21 mars 2023, 08:00 → 23 mars 2023, 18:00 Europe/Paris

Amphi Schwartz (Institut de Mathématiques de Toulouse)

The final conference of the ANR project MESA - Stein's Method and Applications will take place from 21 to 23 March 2023 at the Institute of Mathematics in Toulouse, France.
It aims to give an overview on the use of Stein’s method in probability theory and its applications, with focus on functional inequalities, Malliavin calculus, statistical physics and geometry.

Speakers:

• Benjamin Arras (Laboratoire Paul Painlevé, Université de Lille)
• Bertrand Cloez (INRAE Montpellier)
• Giovanni Conforti (CMAP, Ecole Polytechnique)
• Laurent Decreusefond (LTCI, Télécom Paris)
• Hélène Halconruy (ESILV)
• Ronan Herry (IRMAR, Université de Rennes 1)
• Mahmoud Khabou (IMT, Université de Toulouse)
• Raphaël Lachièze-Rey (MAP5, Université de Paris)
• Ivan Nourdin (Université du Luxembourg)
• Guillaume Poly (IRMAR, Université de Rennes 1)
• Jordan Serres (CREST - ENSAE)
• Anna Paola Todino (Sapienza University of Rome)

 

Organizing committee: François Chapon, Laure Coutin, Max Fathi and Aldéric Joulin

 

Pratical informations: Institut de Mathématiques de Toulouse, Amphi Schwartz, bât. 1R3

 

Sponsors:

                                               

mardi 21 mars ¶

09:30 → 09:45 Conference opening

09:45 → 10:40 Central convergence on Wiener chaoses always implies asymptotic smoothness and C-infinite convergence of densities¶

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 as the convergence of all its derivatives in every L q (R) for all q ∈
[1, +∞] towards
the corresponding derivatives of the Gaussian density. In
particular, these findings
greatly improve the currently known types of convergence which are total
variation
and entropy that were obtained through Malliavin/Stein method.

Joint work with Ronan Herry and Dominique Malicet

Orateur: Guillaume Poly (IRMAR, Université de Rennes 1)

Exposé-Stein-MESA-POLY.pdf

10:45 → 11:15 Coffee break

11:15 → 12:10 Régularité des lois de formes quadratiques en des variables iid : une approche par forme de Dirichlet¶

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 de Sobolev de la densité. Notre approche se base sur une représentation du carré du champ par des variables gaussiennes et un calcul explicite sur les vas gaussiennes. Je présenterai une application à la régularité des de la loi d'une forme quadratique évaluée en une suite de vas iid.
Travail en collaboration avec Dominique Malicet et Guillaume Poly.

Orateur: Ronan Herry (IRMAR, Université de Rennes 1)

12:15 → 14:00 Lunch at l'Esplanade

14:00 → 14:55 Exponential convergence of Sinkhorn algorithm for quadratic entropic optimal transport¶

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 means of Sinkhorn’s algorithm, a.k.a. Iterative Proportional Fitting Procedure. While the exponential convergence of Sinkhorn’s iterates is well understood in a discrete setting or for compactly supported measures and bounded costs, when moving to unbounded costs and non compact marginals the picture is far less clear. In this talk, we shall present an exponential convergence result in the landmark example of quadratic entropic optimal transport and approximately log-concave marginals. The main innovation in the proof strategy are new propagation of weak convexity results along Hamilton Jacobi Bellman equations, that may be of independent interest. Finally, we will highlight how Stein’s method could potentially lead to improvement and extension of our results.

Joint work(s) with Alain Durmus, Giacomo Greco and Maxence Noble

Orateur: Giovanni Conforti (CMAP École Polytechnique)

talkselectrictemplate (58) - conforti.pdf

14:55 → 15:50 Stein's method for stability estimates of the Poincaré constant¶

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 in particular on Stein's method for general one-dimensional distributions.

Orateur: Jordan Serres (CREST - ENSAE)

15:50 → 16:20 Coffee break

16:20 → 17:15 Malliavin calculus for marked binomial processes and Chen-Stein method¶

We can observe a clumping phenomenon when counting the number of series of $t$ heads in a sequence of independent coin tosses or the occurrences of a rare word in a DNA sequence. The Chen-Stein method is an efficient tool to limit the approximation error when the law of the number of clusters can be approximated by a Poisson law (possibly compound).
We revisit this method by reducing these two problems to that of a Poisson approximation for functionals of marked binomial processes (MBPs), which are discrete analogues of marked Poisson processes. We then develop stochastic analysis tools and a Malliavin calculus for MBPs. Under this new formalism, we obtain a general criterion - for the distance in total variation - of the Poisson approximation for MBP functionals, in terms of Malliavin operators. In this talk, I will give elements of the Malliavin formalism for MBPs, state the general result of the approximation and illustrate it by applying it to the two situations of interest.

Orateur: Hélène Halconruy (ESILV)

halconruy.pdf

mercredi 22 mars ¶

09:00 → 09:55 The normal approximation of compound Hawkes functionals¶

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 may not converge to zero in large time, or have been obtained only for specific kernels such as the exponential or Erlang functions.

Orateur: Mahmoud Khabou (IMT, Université de Toulouse)

09:55 → 10:25 Coffee break

10:25 → 11:20 Total variation bound for Hadwiger's functional using Stein's method¶

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 ${\mu }_K(x)=c{e}^{-\pi {\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}^2(x,K)}$, $x\in {\mathbb{R}}^d$. Finally, let the information content $H_K$ be defined as $H_K={\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}^2(X_K,K)$.

In this talk, we will study the fluctuations of $H_K$ around its expectation as the dimension $d$ go to infinity.
Stein's method plays a crucial role in our analysis.
This is joint work with Valentin Garino.

Orateur: Ivan Nourdin (University of Luxembourg)

Exposé Toulouse-Ivan Nourdin.pdf

11:20 → 12:15 Invertibility of functionals of the Poisson process and applications¶

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 excited point process for a given compensator) as the analog to solving an SDE like (1). We then show a Yamada-Watanabe like theorem for weak and strong solutions to the Hawkes problem.
Some relationships are also established between Hawkes processes and directed transport between point processes.

Orateur: Laurent Decreusefond (LTCI, Télécom Paris)

12:15 → 14:00 Lunch at l'Esplanade

14:00 → 19:30 Free afternoon

19:30 → 22:55 Dinner at Du Plaisir à la Toque restaurant

jeudi 23 mars ¶

09:45 → 10:40 Second order Poincaré inequalities and applications to geometric functionals¶

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.

Orateur: Raphaël Lachièze-Rey (MAP5, Université de Paris)

Talk toulouse - lachieze-rey.pdf

10:40 → 11:10 Coffee break

11:10 → 12:05 Quantitative Generalized CLT with Self-Decomposable Limiting Laws by Spectral Methods¶

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.

Orateur: Benjamin Arras (Laboratoire Paul Painlevé, Université de Lille)

12:15 → 14:00 Lunch at l'Esplanade

14:00 → 14:55 Intertwinings and Stein's magic factors for birth-death processes¶

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.

Orateur: Bertrand Cloez (INRAE Montepellier)

14:55 → 15:50 Central Limit Theorems for Poisson Random Waves¶

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 divergence of eigenvalues and Poisson governing measures. The results were obtained exploiting Stein-Malliavin techniques on the Poisson space for the univariate and the multivariate case.

Orateur: Anna Paola Todino (Università degli Studi di Milano-Bicocca)

Todino-Toulouse.pdf

15:50 → 16:20 Coffee break