Limit theorems for anisotropic functionals of stationary Gaussian fields with Gneiting covariance function
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908871627374592 |
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| author | Leonenko, Nikolai Maini, Leonardo Nourdin, Ivan Pistolato, Francesca |
| author_facet | Leonenko, Nikolai Maini, Leonardo Nourdin, Ivan Pistolato, Francesca |
| contents | We study non-linear additive functionals of stationary Gaussian fields over anisotropically growing domains in $\mathbb{R}^d$, including spatiotemporal settings, and establish Gaussian and non-Gaussian limit theorems under non-separable covariance structures. We characterize the regimes in which the normalized functionals converge either to a Gaussian distribution or to a $2$-domain Rosenblatt distribution, depending on precise long-range dependence conditions. Our analysis covers covariance functions from the Gneiting class, which provides a canonical family of non-separable spatiotemporal models. A key structural result shows that such covariances are asymptotically separable in a precise cumulant sense, allowing us to identify explicitly the limiting distributions without imposing additional spectral assumptions. These results extend existing spatiotemporal limit theorems beyond separable and short-memory frameworks and provide a unified description of anisotropic long-range dependence phenomena. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07188 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Limit theorems for anisotropic functionals of stationary Gaussian fields with Gneiting covariance function Leonenko, Nikolai Maini, Leonardo Nourdin, Ivan Pistolato, Francesca Probability Primary 60F05, secondary 60G15, 60G60, 60J55 We study non-linear additive functionals of stationary Gaussian fields over anisotropically growing domains in $\mathbb{R}^d$, including spatiotemporal settings, and establish Gaussian and non-Gaussian limit theorems under non-separable covariance structures. We characterize the regimes in which the normalized functionals converge either to a Gaussian distribution or to a $2$-domain Rosenblatt distribution, depending on precise long-range dependence conditions. Our analysis covers covariance functions from the Gneiting class, which provides a canonical family of non-separable spatiotemporal models. A key structural result shows that such covariances are asymptotically separable in a precise cumulant sense, allowing us to identify explicitly the limiting distributions without imposing additional spectral assumptions. These results extend existing spatiotemporal limit theorems beyond separable and short-memory frameworks and provide a unified description of anisotropic long-range dependence phenomena. |
| title | Limit theorems for anisotropic functionals of stationary Gaussian fields with Gneiting covariance function |
| topic | Probability Primary 60F05, secondary 60G15, 60G60, 60J55 |
| url | https://arxiv.org/abs/2603.07188 |