Diffusion approximations via Stein's method and time changes

Abstract

We extend the ideas of (Barbour 1990) and use Stein's method to obtain a bound on the distance between a scaled time-changed random walk and a time-changed Brownian Motion. We then apply this result to bound the distance between a time-changed compensated scaled Poisson process and a time-changed Brownian Motion. This allows us to bound the distance between a process whose dynamics resemble those of the Moran model with mutation and a process whose dynamics resemble those of the Wright-Fisher diffusion with mutation upon noting that the former may be expressed as a difference of two time-changed Poisson processes and the diffusive part of the latter may be expressed as a time-changed Brownian Motion. The method is applicable to a much wider class of examples satisfying the Stroock-Varadhan theory of diffusion approximation.

Cited by in corpus (5)

• Stein's method for multivariate Brownian approximations of sums under dependence
• Donsker's theorem in {Wasserstein}-1 distance
• Functional approximations with Stein's method of exchangeable pairs
• Diffusive limits of Lipschitz functionals of Poisson measures
• Stein's method for diffusive limit of queueing processes