Threshold for the expected measure of the convex hull of random points with independent coordinates

Abstract

Let $\mu$ be an even Borel probability measure on $\mathbb{R}$. For every $N>n$ consider $N$ independent random vectors ${\vec{X}}_1,\dots ,{\vec{X}}_N$ in ${\mathbb{R}}^n$, with independent coordinates having distribution $\mu$. We establish a sharp threshold for the product measure ${\mu }_n$ of the random polytope $K_N:=\mathrm{conv}\{{\vec{X}}_1,\dots ,{\vec{X}}_N\}$ in ${\mathbb{R}}^n$ under the assumption that the Legendre transform ${\Lambda }_{\mu }^{\ast }$ of the logarithmic moment generating function of $\mu$ satisfies the condition $$\underset{x\uparrow x^{\ast }}{\lim }\frac{-\ln \mu ([x,\infty ))}{{\Lambda }_{\mu }^{\ast }(x)}=1,$$ where $x^{\ast }=\sup \{x\in \mathbb{R}:\mu ([x,\infty ))>0\}$. An application is a sharp threshold for the case of the product measure ${\nu }_p^n={\nu }_p^{\otimes n}$, $p\ge 1$ with density $(2{\gamma }_p{)}^{-n}\exp (-\Vert x{\Vert }_p^p)$, where $\Vert \cdot {\Vert }_p$ is the ${\ell }_p^n$-norm and ${\gamma }_p=\Gamma (1+1/p)$.

References in corpus (1)

• Threshold for the expected measure of random polytopes