Threshold for the volume spanned by random points with independent coordinates
type: Item
instance of: scholarly article
title: Threshold for the volume spanned by random points with independent coordinates
zbMATH Open document ID: 1166.52009
published in: Israel Journal of Mathematics
publication date: 2009-02-17T00:00:00Z
review text: Let \(\mu\) be an even Borel probability measure on the real line \(\mathbb{R}\) with compact support, and define \(\alpha := \sup\{x \in \mathbb{R} : \mu([x,\infty)) > 0\}\) to be the right end-point of the support of \(\mu\). Let \(X\) be a random variable with distribution \(\mu\), let \(\varphi(t) := E(e^{tX})\) denote the moment generating function of \(X\), and let \(\psi(t) := \ln\varphi(t)\) be its cumulant generating function. Let \[ \lambda(x) := \sup\{tx - \psi(t) : t \in \mathbb{R}\} \] be the Legendre transform of \(\psi\), and define \[ \kappa := \frac1{2\alpha} \int_{-\alpha}^\alpha \, \lambda(x)\,dx. \] Now let \(X_1,\ldots,X_n\) be independent and identically distributed random variables with distribution \(\mu\), set \(\mathbf{X} = (X_1,\ldots,X_n)\) and, for fixed \(N > n\) take \(N\) independent copies \(\mathbf{X}_1,\ldots,\mathbf{X}_N\) of \(\mathbf{X}\); this defines the random polytope \(K_N := {\text{conv}}\{\mathbf{X}_1,\ldots,\mathbf{X}_N\}\). If \[ \lim_{x \uparrow \alpha} \, \frac{-\ln\mu([x,\infty))}{\lambda(x)} = 1, \] then the authors show that, for every \(\varepsilon > 0\), the expected volume of \(K_N\) satisfies \[ \begin{aligned} \lim_{n \to \infty} \sup\{(2\alpha)^{-n}E(|K_N|) : N \leq \exp((\kappa - \varepsilon)n)\} &= 0, \\ \lim_{n \to \infty} \inf\{(2\alpha)^{-n}E(|K_N|) : N \geq \exp((\kappa + \varepsilon)n)\} &= 1. \end{aligned} \]
reviewed by: Peter McMullen
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 5508737
zbMATH Keywords: Borel probability measure
zbMATH Keywords: random polytope
zbMATH Keywords: volume
zbMATH Keywords: threshold
author: Dimitris Gatzouras
author: Apostolos Giannopoulos
MaRDI profile type: Publication
full work available at URL: s11856-009-0007-z
OpenAlex ID: W2113331872
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DOI: 10.1007/S11856-009-0007-Z
description: scientific article; zbMATH DE number 5508737