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

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.

accepted for publication in Stochastic Processes and Their Applications

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• Normal approximation for nonlinear statistics using a concentration inequality approach
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• The multivariate functional de Jong CLT
• Malliavin-Stein Method: a Survey of Recent Developments