# 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

## 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

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

Submitted

@unpublished{coutin:hal-02098892,
TITLE = {{Donsker's theorem in {Wasserstein}-1 distance}},
AUTHOR = {Coutin, L. and Decreusefond, Laurent},
URL = {https://hal.telecom-paristech.fr/hal-02098892},
NOTE = {working paper or preprint},
YEAR = {2019},
MONTH = Apr,
KEYWORDS = {Stein method ; Malliavin calculus},
PDF = {https://hal.telecom-paristech.fr/hal-02098892/file/donskerLipschitz.pdf},
HAL_ID = {hal-02098892},
HAL_VERSION = {v1}}


## 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},
}