Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910635757928448 |
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| author | Matsumoto, Tadashi Sullivan, T. J. |
| author_facet | Matsumoto, Tadashi Sullivan, T. J. |
| contents | Gaussian processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP $T u$ that is the image of another GP $u$ under a linear transformation $T$ acting on the sample paths of $u$ are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when $T$ is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator $T$ acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille's theorem for the Bochner integral of a Banach-valued random variable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03594 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators Matsumoto, Tadashi Sullivan, T. J. Probability Statistics Theory 60G12, 60G15, 46G10, 47B01 Gaussian processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP $T u$ that is the image of another GP $u$ under a linear transformation $T$ acting on the sample paths of $u$ are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when $T$ is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator $T$ acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille's theorem for the Bochner integral of a Banach-valued random variable. |
| title | Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators |
| topic | Probability Statistics Theory 60G12, 60G15, 46G10, 47B01 |
| url | https://arxiv.org/abs/2305.03594 |