Open Access
May 2024 A diffusion approach to Stein’s method on Riemannian manifolds
Huiling Le, Alexander Lewis, Karthik Bharath, Christopher Fallaize
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Bernoulli 30(2): 1079-1104 (May 2024). DOI: 10.3150/23-BEJ1625


We detail an approach to developing Stein’s method for bounding integral metrics on probability measures defined on a Riemannian manifold M. Our approach exploits the relationship between the generator of a diffusion on M having a target invariant measure and its characterising Stein operator. We consider a pair of such diffusions with different starting points, and through analysis of the distance process between the pair, derive Stein factors, which bound the solution to the Stein equation and its derivatives. The Stein factors contain curvature-dependent terms and reduce to those currently available for Rm, and moreover imply that the bounds for Rm remain valid when M is a flat manifold.


We are grateful to Dennis Barden for permission to include his unpublished work (Lemma 3 in Appendix A of Supplementary Material), jointly with HL. This result lays the geometric foundation for our analysis of the distance between the pair of diffusions. KB acknowledges partial support from grants EPSRC EP/V048104/1, NSF 2015374 and NIH R37-CA214955. For the purpose of open access, the authors have applied a creative commons attribution (CC BY) licence to any author accepted manuscript version arising.


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Huiling Le. Alexander Lewis. Karthik Bharath. Christopher Fallaize. "A diffusion approach to Stein’s method on Riemannian manifolds." Bernoulli 30 (2) 1079 - 1104, May 2024.


Received: 1 June 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699546
Digital Object Identifier: 10.3150/23-BEJ1625

Keywords: coupling , integral metrics , Stein equation , stochastic flow , Wasserstein distance

Rights: This research was funded, in whole or in part, by [Engineering and Physical Sciences Research Council (EPSRC), UK, EPSRC EP/V048104/1]. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant’s open access conditions

Vol.30 • No. 2 • May 2024
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