# Stein’s method in functional settings¶

## My list of publications about Stein’s method¶

The Wasserstein-1 distance between too probability measures on a space $$E$$ is defined as

$\sup_{F\in \text{Lip}_1(E)} \left(\int_E F \ d\mu-\int_E F\ d\, \nu\right)$

It was realized relatively recently that Malliavin can be greatly helpful to make the necessary computations. For details on the so-called Stein-Dirichlet-Malliavin method, please read this paper

## Diffusive limits of Lipschitz functionals of Poisson measures¶

The Wasserstein-1 distance is by definition, stable with respect to Lipschitz transformations. We show that diffusion approximations can be obtained as such transformations of the Donsker’s theorem for renormalized Poisson process. This yields convergence rate to limiting processes which are not Gaussian.

Note

• ArXiv : 2107.05339

## Donsker theorem in Wasserstein-1 distance¶

Abstract: In previous papers hal-01551694 and hal-00717812, we analyzed the rate of convergence of a random walk

$X^m(t)=\frac{1}{\sqrt{m}}\left( \sum_{j=1}^{[mt]} X_j+(mt-[mt])X_{[mt+1]}\right)$

to a Brownian motion in different functional spaces. We here at last prove that

$\sup_{F\in \text{Lip}_1(W_{\alpha,p})} \left(\mathbf E[F(X^m)]-\mathbf E[F(B)]\right)\le c m^{-1/6+\alpha/3}\log m$

where

$W_{\alpha,p}=\{f\, :\, [0,1]\to \mathbf R,\ \int_0^1\int_0^1 \frac{|f(t)-f(s)|^p}{|t-s|^{1+\alpha p}}\ ds\ dt<\infty\}$

Note

@Article{Coutin2020,
author       = {Coutin, L. and Decreusefond, L.},
date         = {2020},
journaltitle = {Electronic Communications in Probability},
title        = {Donsker's theorem in {W}asserstein-1 distance},
doi          = {10.1214/20-ecp308},
pages        = {Paper No. 27, 13},
volume       = {25},
file         = {:Coutin/Coutin2020.pdf:PDF},
mrnumber     = {4089734}}

## Functional Poisson approximation¶

Abstract: A Poisson or a binomial process on an abstract state space and a symmetric function f acting on k-tuples of its points are considered. They induce a point process on the target space of f. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein’s method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived and examples from stochastic geometry are investigated.

@article{decreusefond:hal-01010967,
TITLE = {{Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry}},
AUTHOR = {Decreusefond, Laurent and Schulte, Matthias and Th{"a}le, Christoph},
URL = {https://hal.archives-ouvertes.fr/hal-01010967},
JOURNAL = {{Annals of Probability}},
VOLUME = {44},
NUMBER = {3},
PAGES = {2147-2197},
YEAR = {2016},
KEYWORDS = {Poisson process ; Stein method ; stochastic geometry ; Malliavin calculus},
PDF = {https://hal.archives-ouvertes.fr/hal-01010967/file/PPPConv13.pdf},
HAL_ID = {hal-01010967},
}