Functional approximations with Stein's method of exchangeable pairs

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.

will appear in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

References in corpus (6)

• Exchangeable pairs and Poisson approximation
• Exponential Approximation by Stein's Method and Spectral Graph Theory
• Stein's method for multivariate Brownian approximations of sums under dependence
• Diffusion approximations via Stein's method and time changes
• Note on A. Barbour's paper on Stein's method for diffusion approximations
• Stein's method for diffusive limit of queueing processes

Cited by in corpus (4)

• Stein's method for multivariate Brownian approximations of sums under dependence
• Stein's method, Gaussian processes and Palm measures, with applications to queueing
• Stein's method, smoothing and functional approximation
• The multivariate functional de Jong CLT