Stein's method for multivariate Brownian approximations of sums under dependence

Mikołaj Kasprzak

Published 2017-08-08Version 1

We use Stein's method to obtain a bound on the distance between scaled $p$-dimensional random walks and a $p$-dimensional (correlated) Brownian Motion. We consider dependence schemes including those in which the summands in scaled sums are weakly dependent and their $p$ components are strongly correlated. We also find a bound on the rate of convergence of scaled U-statistics to Brownian Motion, representing an example of a sum of strongly dependent terms. Furthermore, we consider an $r$-regular dependency graph with independent compensated Poisson processes assigned to its edges. We calculate the distance between the $p$-dimensional vector of sums of those processes attached to $p$ vertices and a $p$-dimensional standard Brownian Motion.