Threshold for the expected measure of random polytopes

Abstract

Let $μ$ be a log-concave probability measure on ${\mathbb R}^n$ and for any $N>n$ consider the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$, where $X_1,X_2,\ldots $ are independent random points in ${\mathbb R}^n$ distributed according to $μ$. We study the question if there exists a threshold for the expected measure of $K_N$. Our approach is based on the Cramer transform $Λ_μ^{\ast }$ of $μ$. We examine the existence of moments of all orders for $Λ_μ^{\ast }$ and establish, under some conditions, a sharp threshold for the expectation ${\mathbb E}_{μ^N}[μ(K_N)]$ of the measure of $K_N$: it is close to $0$ if $\ln N\ll {\mathbb E}_{μ}(Λ_μ^{\ast })$ and close to $1$ if $\ln N\gg {\mathbb E}_{μ}(Λ_μ^{\ast })$. The main condition is that the parameter $β(μ)={\rm Var}_{μ}(Λ_μ^{\ast })/({\mathbb E}_{μ}(Λ_{μ}^{\ast }))^2$ should be small.