Abstract
It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of the samples of a Gaussian process X is controlled by a certain nuclear dominance condition. However, it is less clear how to identify a “small” set of functions (not necessarily a vector space) that contains the samples. This article presents a general approach for identifying such sets. We use scaled RKHSs, which can be viewed as a generalisation of Hilbert scales, to define the sample support set as the largest set which is contained in every element of full measure under the law of X in the σ-algebra induced by the collection of scaled RKHS. This potentially non-measurable set is then shown to consist of those functions that can be expanded in terms of an orthonormal basis of the RKHS of the covariance kernel of X and have their squared basis coefficients bounded away from zero and infinity, a result suggested by the Karhunen–Loève theorem.
Funding Statement
This work was supported by the Lloyd’s Register Foundation Programme for Data-Centric Engineering at the Alan Turing Institute, United Kingdom and the Academy of Finland postdoctoral researcher grant #338567 “Scalable, adaptive and reliable probabilistic integration”.
Acknowledgements
I am grateful to Motonobu Kanagawa for his patience in enduring persistent questions over the course of a conference in Sydney in July 2019 during which the main ideas underlying this article were first conceived and to Chris Oates for extensive commentary on the draft at various stages of preparation. Discussions with Anatoly Zhigljavsky led to improvements in Section 6.
Citation
Toni Karvonen. "Small sample spaces for Gaussian processes." Bernoulli 29 (2) 875 - 900, May 2023. https://doi.org/10.3150/22-BEJ1483