Contributions to Stein's method on Riemannian manifolds

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