Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2308.04242 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913577493856256 |
|---|---|
| author | Visonà, Tommaso |
| author_facet | Visonà, Tommaso |
| contents | Let $Ξ_n=\{ξ_1,\dots,ξ_n\}$ be a sample of $n$ independent points distributed in a regular closed element $K$ of the extended convex ring in $\mathbb{R}^d$ according to a probability measure $μ$ on $K$, admitting a density function. We consider random sets generated from the intersection of the translations of $K$ by elements of $Ξ_n$, as $X_n=\bigcap_{i=1}^n (K-ξ_i)$. This work aims to show that the scaled closure of the complement of $X_n$ as $n\to\infty$ converges in distribution to the closure of the complement zero cell of a Poisson hyperplane tessellation whose distribution is determined by the curvature measure of $K$ and the behaviour of the density of $μ$ near the boundary of $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_04242 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Intersections of randomly translated sets Visonà, Tommaso Probability 60D05 60G55 52A22 Let $Ξ_n=\{ξ_1,\dots,ξ_n\}$ be a sample of $n$ independent points distributed in a regular closed element $K$ of the extended convex ring in $\mathbb{R}^d$ according to a probability measure $μ$ on $K$, admitting a density function. We consider random sets generated from the intersection of the translations of $K$ by elements of $Ξ_n$, as $X_n=\bigcap_{i=1}^n (K-ξ_i)$. This work aims to show that the scaled closure of the complement of $X_n$ as $n\to\infty$ converges in distribution to the closure of the complement zero cell of a Poisson hyperplane tessellation whose distribution is determined by the curvature measure of $K$ and the behaviour of the density of $μ$ near the boundary of $K$. |
| title | Intersections of randomly translated sets |
| topic | Probability 60D05 60G55 52A22 |
| url | https://arxiv.org/abs/2308.04242 |