On the orthogonality of zero-mean Gaussian measures: Sufficiently dense sampling
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866909176625627136 |
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| author | Furrer, Reinhard Hediger, Michael |
| author_facet | Furrer, Reinhard Hediger, Michael |
| contents | For a stationary random function $ξ$, sampled on a subset $D$ of $\mathbb{R}^{d}$, we examine the equivalence and orthogonality of two zero-mean Gaussian measures $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ associated with $ξ$. We give the isotropic analog to the result that the equivalence of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ is linked with the existence of a square-integrable extension of the difference between the covariance functions of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ from $D$ to $\mathbb{R}^{d}$. We show that the orthogonality of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ can be recovered when the set of distances from points of $D$ to the origin is dense in the set of non-negative real numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_10239 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the orthogonality of zero-mean Gaussian measures: Sufficiently dense sampling Furrer, Reinhard Hediger, Michael Probability 60G10, 60G15, 60G17, 60G30, 60G60 For a stationary random function $ξ$, sampled on a subset $D$ of $\mathbb{R}^{d}$, we examine the equivalence and orthogonality of two zero-mean Gaussian measures $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ associated with $ξ$. We give the isotropic analog to the result that the equivalence of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ is linked with the existence of a square-integrable extension of the difference between the covariance functions of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ from $D$ to $\mathbb{R}^{d}$. We show that the orthogonality of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ can be recovered when the set of distances from points of $D$ to the origin is dense in the set of non-negative real numbers. |
| title | On the orthogonality of zero-mean Gaussian measures: Sufficiently dense sampling |
| topic | Probability 60G10, 60G15, 60G17, 60G30, 60G60 |
| url | https://arxiv.org/abs/2212.10239 |