Sojourns of Vector-Valued Stationary Gaussian Random Fields
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914081335672832 |
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| author | Dębicki, Krzysztof Hashorva, Enkelejd Michna, Zbigniew |
| author_facet | Dębicki, Krzysztof Hashorva, Enkelejd Michna, Zbigniew |
| contents | For a centered, homogeneous R^d-valued Gaussian random field X(t), t in R^k, with covariance matrix function R(s,t) = E[X(s) X(t)^T], we investigate the exact asymptotics of kappa_u(x) = P( theta(u) * integral over [0,T]^k of 1{X(t) > u b} dt > x ), where b = (b1, ..., bd)^T, as u -> infinity, with x >= 0 and T > 0, and theta(u) is a scaling function related to the expansion of R(s,t) around (0,0).
To approximate kappa_u(x), we extend both Berman's original approach and the uniform double-sum method to the multivariate setting. Furthermore, we derive the exact asymptotics for the supremum of X, thus extending several recent results in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sojourns of Vector-Valued Stationary Gaussian Random Fields Dębicki, Krzysztof Hashorva, Enkelejd Michna, Zbigniew Probability Primary 60G15, secondary 60G70 For a centered, homogeneous R^d-valued Gaussian random field X(t), t in R^k, with covariance matrix function R(s,t) = E[X(s) X(t)^T], we investigate the exact asymptotics of kappa_u(x) = P( theta(u) * integral over [0,T]^k of 1{X(t) > u b} dt > x ), where b = (b1, ..., bd)^T, as u -> infinity, with x >= 0 and T > 0, and theta(u) is a scaling function related to the expansion of R(s,t) around (0,0). To approximate kappa_u(x), we extend both Berman's original approach and the uniform double-sum method to the multivariate setting. Furthermore, we derive the exact asymptotics for the supremum of X, thus extending several recent results in the literature. |
| title | Sojourns of Vector-Valued Stationary Gaussian Random Fields |
| topic | Probability Primary 60G15, secondary 60G70 |
| url | https://arxiv.org/abs/2510.06930 |