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...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
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2023
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| Online Access: | https://eprints.nottingham.ac.uk/72476/ |
| _version_ | 1848800743897694208 |
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| author | Lewis, Alexander |
| author_facet | Lewis, Alexander |
| author_sort | Lewis, Alexander |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:56:25Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-72476 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:56:25Z |
| publishDate | 2023 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-724762023-07-26T04:40:16Z https://eprints.nottingham.ac.uk/72476/ Contributions to Stein's method on Riemannian manifolds Lewis, Alexander 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. 2023-07-26 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en cc_by https://eprints.nottingham.ac.uk/72476/1/Alexander%20Lewis%20PhD%20Thesis.pdf Lewis, Alexander (2023) Contributions to Stein's method on Riemannian manifolds. PhD thesis, University of Nottingham. Stein's method Riemannian geometry probability theory stochastic differential equations Wasserstein metric |
| spellingShingle | Stein's method Riemannian geometry probability theory stochastic differential equations Wasserstein metric Lewis, Alexander Contributions to Stein's method on Riemannian manifolds |
| title | Contributions to Stein's method on Riemannian manifolds |
| title_full | Contributions to Stein's method on Riemannian manifolds |
| title_fullStr | Contributions to Stein's method on Riemannian manifolds |
| title_full_unstemmed | Contributions to Stein's method on Riemannian manifolds |
| title_short | Contributions to Stein's method on Riemannian manifolds |
| title_sort | contributions to stein's method on riemannian manifolds |
| topic | Stein's method Riemannian geometry probability theory stochastic differential equations Wasserstein metric |
| url | https://eprints.nottingham.ac.uk/72476/ |