# 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

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`

### The latest publications¶

- Read more about Donsker’s theorem in Wassersten-1 distance.
- Read more about Functional Poisson approximation.

## 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

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

where

*Submitted*

Note

```
@unpublished{coutin:hal-02098892,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}, }