KMT coupling for random walk bridges

schedule le lundi 15 juin 2020 de 17h00 à 18h00

Organisé par : F. Bechtold, W. Da Silva , A. Fermanian, S. Has, Y. Yu

Intervenant : Xuan Wu (http://www.math.columbia.edu/~xuanw/) (Columbia University)
Lieu : Online at https://bigbluebutton.math.upmc.fr/b/ade-phf-9dg

Sujet : KMT coupling for random walk bridges

Résumé :

A classical problem in probability theory, called the embedding problem, asks to construct a random walk process $S_n = X_1+...+X_n$ ($X_i$ are i.i.d) and a standard Brownian motion $B_t$ on thesame probability space so that the uniform distance max_{1 <= k <= n}|S_k − B_k| grows as slowly as possible in $n$. Komlos, Major and Tusnady showed that one can achieve max_{1 <= k <= n}|S_k âˆ’ B_k| = O(log n) for the rate of strong coupling, provided that the jump distribution $X$ has a finite moment generating function in a neighborhood of zero in 1970s. The construction used to achieve this celebrated result is now referred to as the KMT approximation or coupling. Unless $X$ is normally distributed, the $log n$ rate of approximation is optimal. Since its inception, the KMT coupling has become an invaluable tool in probability theory and statistics. In this talk, we will investigate on a generalization to the setting of random walk bridges, which requires more refined estimates and stronger assumptions necessitated by the endpoint conditioning. In particular, our result does not follow from the KMT embedding theorem, which we illustrate via a counterexample. This talk is based on a joint work with Evgeni Dimitrov.