On the moments of the volume for random convex chains
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909852928835584 |
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| author | Gusakova, Anna Muranova, Anna |
| author_facet | Gusakova, Anna Muranova, Anna |
| contents | Let $T$ be the triangle in the plane with vertices $(0, 0)$, $(0,1)$ and $(0, 1)$. The convex hull $T_n$ of points $(0, 1)$, $(1, 0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain. In this paper we study the moments of the volume of random polytope $T_n$ and derive exact formulas for $k$-th moments for any integer $k\ge 0$. As an intermediate result, we find an explicit representation for the probability generating function of the number of vertices of $T_n$, from which an alternative formula for the probability that $T_n$ has $k$ vertices follows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15807 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the moments of the volume for random convex chains Gusakova, Anna Muranova, Anna Probability Combinatorics 52A22, 60D05, 33C05, 05A15 Let $T$ be the triangle in the plane with vertices $(0, 0)$, $(0,1)$ and $(0, 1)$. The convex hull $T_n$ of points $(0, 1)$, $(1, 0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain. In this paper we study the moments of the volume of random polytope $T_n$ and derive exact formulas for $k$-th moments for any integer $k\ge 0$. As an intermediate result, we find an explicit representation for the probability generating function of the number of vertices of $T_n$, from which an alternative formula for the probability that $T_n$ has $k$ vertices follows. |
| title | On the moments of the volume for random convex chains |
| topic | Probability Combinatorics 52A22, 60D05, 33C05, 05A15 |
| url | https://arxiv.org/abs/2510.15807 |