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

Mikolaj Jakub Kasprzak

Abstract: We use Stein’s method to obtain a bound on the distance between scaled p-dimensional random walks and a -dimensional (correlated) Brownian motion. We consider dependence schemes including those in which the summands in scaled sums are weakly dependent and their components are strongly correlated. As an example application, we prove a functional limit theorem for exceedances in an m-scans process, together with a bound on the rate of convergence. 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.