Functional approximations with Stein's method of exchangeable pairs

TL;DR

This paper introduces a novel approach combining exchangeable pairs with Stein's method to achieve functional Gaussian approximations for stochastic processes, including combinatorial CLTs and graph processes.

Contribution

It provides a general linearity condition for Gaussian approximation of stochastic processes and applies it to combinatorial and graph-valued processes.

Findings

Established a general Gaussian approximation theorem for stochastic processes.

Proved a functional combinatorial central limit theorem.

Bounded the convergence speed of graph process edge counts to Gaussian processes.

Abstract

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