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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.17538 |
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| _version_ | 1866909160317124608 |
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| author | Caponera, Alessia Rossi, Maurizia Medina, María Dolores Ruiz |
| author_facet | Caponera, Alessia Rossi, Maurizia Medina, María Dolores Ruiz |
| contents | In this note we investigate geometric properties of invariant spatio-temporal random fields $X:\mathbb M^d\times \mathbb R\to \mathbb R$ defined on a compact two-point homogeneous space $\mathbb M^d$ in any dimension $d\ge 2$, and evolving over time. In particular, we focus on chi-squared distributed random fields, and study the large time behavior (as $T\to +\infty$) of the average on $[0,T]$ of the volume of the excursion set on the manifold, i.e., of $\lbrace X(\cdot, t)\ge u\rbrace$ (for any $u >0$). The Fourier components of $X$ may have short or long memory in time, i.e., integrable or non-integrable temporal covariance functions. Our argument follows the approach developed in (Marinucci, Rossi, Vidotto (2021) Ann. Appl. Probab.) and allow to extend their results for invariant spatio-temporal Gaussian fields on the two-dimensional unit sphere to the case of chi-squared distributed fields on two-point homogeneous spaces in any dimension. We find that both the asymptotic variance and limiting distribution, as $T\to +\infty$, of the average empirical volume turn out to be non-universal, depending on the memory parameters of the field $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17538 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sojourn functionals of time-dependent $χ^2$-random fields on two-point homogeneous spaces Caponera, Alessia Rossi, Maurizia Medina, María Dolores Ruiz Probability In this note we investigate geometric properties of invariant spatio-temporal random fields $X:\mathbb M^d\times \mathbb R\to \mathbb R$ defined on a compact two-point homogeneous space $\mathbb M^d$ in any dimension $d\ge 2$, and evolving over time. In particular, we focus on chi-squared distributed random fields, and study the large time behavior (as $T\to +\infty$) of the average on $[0,T]$ of the volume of the excursion set on the manifold, i.e., of $\lbrace X(\cdot, t)\ge u\rbrace$ (for any $u >0$). The Fourier components of $X$ may have short or long memory in time, i.e., integrable or non-integrable temporal covariance functions. Our argument follows the approach developed in (Marinucci, Rossi, Vidotto (2021) Ann. Appl. Probab.) and allow to extend their results for invariant spatio-temporal Gaussian fields on the two-dimensional unit sphere to the case of chi-squared distributed fields on two-point homogeneous spaces in any dimension. We find that both the asymptotic variance and limiting distribution, as $T\to +\infty$, of the average empirical volume turn out to be non-universal, depending on the memory parameters of the field $X$. |
| title | Sojourn functionals of time-dependent $χ^2$-random fields on two-point homogeneous spaces |
| topic | Probability |
| url | https://arxiv.org/abs/2403.17538 |