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

Diffusion approximations via Stein's method and time changes