Karhunen-Loève expansion of Random Measures
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866909653119533056 |
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| author | Vergara, Ricardo Carrizo |
| author_facet | Vergara, Ricardo Carrizo |
| contents | We present an orthogonal expansion for real, function-regulated, second-order random measures over $\mathbb{R}^{d}$ with measure covariance. Such a expansion, which can be seen as a Karhunen-Loève decomposition, consists in a series of deterministic real measures weighted by uncorrelated real random variables with the variances forming a convergent series. The convergence of the series is in a mean-square sense stochastically and against measurable bounded test functions (with compact support if the random measure is not finite) in the measure sense, which implies set-wise convergence. This is proven taking advantage of the extra requirement of having a covariance measure over $\mathbb{R}^{d}\times\mathbb{R}^{d}$ describing the covariance structure of the random measure, for which we also provide a series expansion. These results cover for instance the cases of Gaussian White Noise, Poisson and Cox point processes, and can be used to obtain expansions for trawl processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_14202 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Karhunen-Loève expansion of Random Measures Vergara, Ricardo Carrizo Probability Functional Analysis Statistics Theory 60G57, 60G12, 60H05 We present an orthogonal expansion for real, function-regulated, second-order random measures over $\mathbb{R}^{d}$ with measure covariance. Such a expansion, which can be seen as a Karhunen-Loève decomposition, consists in a series of deterministic real measures weighted by uncorrelated real random variables with the variances forming a convergent series. The convergence of the series is in a mean-square sense stochastically and against measurable bounded test functions (with compact support if the random measure is not finite) in the measure sense, which implies set-wise convergence. This is proven taking advantage of the extra requirement of having a covariance measure over $\mathbb{R}^{d}\times\mathbb{R}^{d}$ describing the covariance structure of the random measure, for which we also provide a series expansion. These results cover for instance the cases of Gaussian White Noise, Poisson and Cox point processes, and can be used to obtain expansions for trawl processes. |
| title | Karhunen-Loève expansion of Random Measures |
| topic | Probability Functional Analysis Statistics Theory 60G57, 60G12, 60H05 |
| url | https://arxiv.org/abs/2203.14202 |