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Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators.
- Source :
-
Analysis & Applications . Apr2024, Vol. 22 Issue 3, p619-633. 15p. - Publication Year :
- 2024
-
Abstract
- 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 02195305
- Volume :
- 22
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Analysis & Applications
- Publication Type :
- Academic Journal
- Accession number :
- 176685697
- Full Text :
- https://doi.org/10.1142/S0219530524400025