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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2405.16461 |
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| _version_ | 1866912422942474240 |
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| author | Penrose, Mathew D. Yang, Xiaochuan |
| author_facet | Penrose, Mathew D. Yang, Xiaochuan |
| contents | Consider a spherical Poisson Boolean model $Z$ in Euclidean $d$-space with $d \geq 2$, with Poisson intensity $t$ and radii distributed like $rY$ with $r \geq 0$ a scaling parameter and $Y$ a fixed nonnegative random variable with finite $(2d-2)$-nd moment (or if $d=2$, a finite $(2 + \varepsilon)$-moment condition for some $\varepsilon >0$). Let $A \subset {\bf R}^d$ be compact with a nice boundary. Let $α$ be the expected volume of a ball of radius $Y$, and suppose $r=r(t)$ is chosen so that $αt r^d - \log t - (d-1) \log \log t$ is a constant independent of $t$. A classical result of Hall and of Janson determines the (non-trivial) large-$t$ limit of the probability that $A$ is fully covered by $Z$. In this paper we provide an $O((\log \log t)/\log t)$ bound on the rate of convergence in that result. With a slight adjustment to $r(t)$, this can be improved to $O(1/\log t)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16461 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the rate of convergence in the Hall-Janson coverage theorem Penrose, Mathew D. Yang, Xiaochuan Probability 60D05 Consider a spherical Poisson Boolean model $Z$ in Euclidean $d$-space with $d \geq 2$, with Poisson intensity $t$ and radii distributed like $rY$ with $r \geq 0$ a scaling parameter and $Y$ a fixed nonnegative random variable with finite $(2d-2)$-nd moment (or if $d=2$, a finite $(2 + \varepsilon)$-moment condition for some $\varepsilon >0$). Let $A \subset {\bf R}^d$ be compact with a nice boundary. Let $α$ be the expected volume of a ball of radius $Y$, and suppose $r=r(t)$ is chosen so that $αt r^d - \log t - (d-1) \log \log t$ is a constant independent of $t$. A classical result of Hall and of Janson determines the (non-trivial) large-$t$ limit of the probability that $A$ is fully covered by $Z$. In this paper we provide an $O((\log \log t)/\log t)$ bound on the rate of convergence in that result. With a slight adjustment to $r(t)$, this can be improved to $O(1/\log t)$. |
| title | On the rate of convergence in the Hall-Janson coverage theorem |
| topic | Probability 60D05 |
| url | https://arxiv.org/abs/2405.16461 |