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

TL;DR

This paper applies Stein's method to derive bounds on the convergence rate of multivariate dependent sums to correlated Brownian motion, with applications to scan processes and U-statistics.

Contribution

It introduces new bounds for multivariate dependent sums approximating Brownian motion, including for strongly dependent components and specific applications.

Findings

Bound on the distance between scaled sums and Brownian motion.

Functional limit theorem for exceedances in m-scans process.

Rate of convergence bounds for scaled U-statistics.

Abstract

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. 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.