On a general Kac-Rice formula for the measure of a level set
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916702455857152 |
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| author | Armentano, Diego Azaïs, Jean-Marc León, José Rafael |
| author_facet | Armentano, Diego Azaïs, Jean-Marc León, José Rafael |
| contents | Let $X(\cdot) $ be a random field $\mathbb{R}^D \to \mathbb{R}^d$, $D\geq d$. We first studied the level set $X^{-1}( u) $, $u \in \mathbb{R}^d$. In particular we gave a weak condition for this level set to be rectifiable. Then, we established a Kac-Rice formula to compute the $D-d$ Hausdorff measure. Our results extend known results, particularly in the non-Gaussian case where we obtained a very general result. We conclude with several extensions and examples of application, including functions of Gaussian random field, zeroes of the likelihood, gravitational microlensing, shot-noise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07424 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a general Kac-Rice formula for the measure of a level set Armentano, Diego Azaïs, Jean-Marc León, José Rafael Probability 60GG60, 60GG15, 60GG10 Let $X(\cdot) $ be a random field $\mathbb{R}^D \to \mathbb{R}^d$, $D\geq d$. We first studied the level set $X^{-1}( u) $, $u \in \mathbb{R}^d$. In particular we gave a weak condition for this level set to be rectifiable. Then, we established a Kac-Rice formula to compute the $D-d$ Hausdorff measure. Our results extend known results, particularly in the non-Gaussian case where we obtained a very general result. We conclude with several extensions and examples of application, including functions of Gaussian random field, zeroes of the likelihood, gravitational microlensing, shot-noise. |
| title | On a general Kac-Rice formula for the measure of a level set |
| topic | Probability 60GG60, 60GG15, 60GG10 |
| url | https://arxiv.org/abs/2304.07424 |