Threshold for the volume spanned by random points with independent coordinates (Q1001443)

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 \[ \begin{aligned} \lim_{n \to \infty} \sup\{(2\alpha)^{-n}E(|K_N|) : N \leq \exp((\kappa - \varepsilon)n)\} &= 0, \\ \lim_{n \to \infty} \inf\{(2\alpha)^{-n}E(|K_N|) : N \geq \exp((\kappa + \varepsilon)n)\} &= 1. \end{aligned} \]