Stein's Method and Analysis – MESA

The approach considered consists in leveraging tools from mathematical analysis (calculus of variations, PDE, functional analysis) in the context of Stein's method.

Results obtained at this time include the development of higher-order Stein kernels (existence, functional inequalities, improved rates of convergence in the CLT), existence of free Stein kernels, new results on stability of Gaussian functional inequalities, a proof of efficiency of the shrinkage estimator in a non-gaussian setting, and rates of convergence for large random matrices with exchangeable blocs.

Ongoing work includes extending the techniques for functional inequalities in a geometric setting, new free quantitative CLT for variables with symmetries and developing new goodness-of-fit tests for random samples

Porject members have published 9 articles in international peer-reviewed journals since the launch, and 14 preprints have been submitted.

The problem of estimating distances between probbaility measures arises in many areas of science (physics, computer science, biology,...), to quantify how accurately some phenomenon is approximated by a given stochastic model. Stein's method is a set of techniques for estimating such distances via well-chosen integration by parts formulas. Since it was first introduced by C. Stein in the 70s, it has found many applications, in areas including quantitative central limit theorems, statistical physics and combinatorics.

This project aims at developing new techniques and applications, by leveraging ideas from mathematical analysis (PDE techniques, variational methods, optimal transport, functional analysis). Fields of application we shall explore include statistics, random matrix theory and free probability, stochastic processes, high-dimensional geometry, statistical physics and Markov Chain Monte Carlo algorithms.