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February, 1982 On a Conjecture in Geometric Probability Regarding Asymptotic Normality of a Random Simplex
A. M. Mathai
Ann. Probab. 10(1): 247-251 (February, 1982). DOI: 10.1214/aop/1176993929

Abstract

A conjecture in geometric probability about the asymptotic normality of the $r$-content of the $r$-simplex, whose $r + 1$ vertices are independently uniformly distributed random points of which $p$ are in the interior and $r + 1 - p$ are on the boundary of a unit $n$-ball, is proved by Ruben (1977). In this article it is shown that the exact density of the random $r$-content is available in the most general case. The technique of inverse Mellin transform is used to get the exact density, thus requiring the knowledge of the $k$th moment of the $r$-content for all real $k$. This $k$th moment is already available in the literature. Approximations and asymptotic results as well as a simpler alternate proof for the conjecture are also given.

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A. M. Mathai. "On a Conjecture in Geometric Probability Regarding Asymptotic Normality of a Random Simplex." Ann. Probab. 10 (1) 247 - 251, February, 1982. https://doi.org/10.1214/aop/1176993929

Information

Published: February, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0479.60021
MathSciNet: MR637392
Digital Object Identifier: 10.1214/aop/1176993929

Subjects:
Primary: 60D05
Secondary: 33A35

Keywords: asymptotic normality , exact density , Random volumes

Rights: Copyright © 1982 Institute of Mathematical Statistics

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Vol.10 • No. 1 • February, 1982
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