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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.13340 |
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| _version_ | 1866909704963227648 |
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| author | Biermé, Hermine Durieu, Olivier Surgailis, Donatas |
| author_facet | Biermé, Hermine Durieu, Olivier Surgailis, Donatas |
| contents | We study the limit distribution of the volume fraction estimator $\widehat p_{λ, A}$ (= the Lebesgue measure of the intersection $\mathcal{X}\cap (λA)$ of a random set $\mathcal{X}$ with a large observation set $λA$, divided by the Lebesgue measure of $λA$), as $λ\to \infty$, for a Boolean set $\mathcal{X}$ formed by uniformly scattered random grains $Ξ\subset \mathbb{R}^ν$. We obtain general conditions on generic grain set $Ξ$ under which $\widehat p_{λ, A}$ has an $α$-stable limit distribution with index $1 < α\le 2$. A large class of Boolean models with randomly homothetic grains satisfying these conditions is introduced. We also discuss the limit distribution of the sample volume fraction of a Boolean set
observed on a large subset of a $ν_0$-dimensional $(1 \le ν_0 \le ν-1$) hyperplane of $\mathbb{R}^ν$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13340 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limit distribution of the sample volume fraction of Boolean set Biermé, Hermine Durieu, Olivier Surgailis, Donatas Probability We study the limit distribution of the volume fraction estimator $\widehat p_{λ, A}$ (= the Lebesgue measure of the intersection $\mathcal{X}\cap (λA)$ of a random set $\mathcal{X}$ with a large observation set $λA$, divided by the Lebesgue measure of $λA$), as $λ\to \infty$, for a Boolean set $\mathcal{X}$ formed by uniformly scattered random grains $Ξ\subset \mathbb{R}^ν$. We obtain general conditions on generic grain set $Ξ$ under which $\widehat p_{λ, A}$ has an $α$-stable limit distribution with index $1 < α\le 2$. A large class of Boolean models with randomly homothetic grains satisfying these conditions is introduced. We also discuss the limit distribution of the sample volume fraction of a Boolean set observed on a large subset of a $ν_0$-dimensional $(1 \le ν_0 \le ν-1$) hyperplane of $\mathbb{R}^ν$. |
| title | Limit distribution of the sample volume fraction of Boolean set |
| topic | Probability |
| url | https://arxiv.org/abs/2505.13340 |