Multivariate functional approximations with Stein's method of exchangeable pairs

Mikolaj J. Kasprzak

Published 2017-10-25Version 1

We combine the multivariate method of exchangeable pairs with Stein's method for functional approximation and give a general linearity condition under which an abstract Gaussian approximation theorem for stochastic processes holds. We apply this approach to estimate the distance from a pre-limiting mixture process of a sum of random variables chosen from an array according to a random permutation and prove a functional combinatorial central limit theorem. We also consider a graph-valued process and bound the speed of convergence of the joint distribution of its rescaled edge and two-star counts to a two-dimensional continuous Gaussian process.