| Summary: | The overarching theme of this thesis is the study of Stein's method on manifolds. We detail an adaptation of the density method on intervals in $\mathbb{R}$ to the unit circle $\mathbb{S}^1$ and give examples of bounds between circular probability distributions. We also use a recently proposed framework to bound the Wasserstein distance between a number of probability measures on Riemannian manifolds with both positive and negative curvature. Particularly, a finite parameter bound on the Wasserstein metric is given between the Riemannian-Gaussian distribution and heat kernel on $\mathbb{H}^3$, which gives a finite sample bound of the Varadhan asymptotic relation in this instance. We then develop a new framework to extend Stein's method to probability measures on manifolds with a boundary. This is done by the addition of a local time term in the diffusion. We find that many results carry over, with appropriate modifications, from the boundary-less case.
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