论文标题
预期的随机多型量度的阈值
Threshold for the expected measure of random polytopes
论文作者
论文摘要
令$μ$为$ {\ mathbb r}^n $上的对数符号的概率度量，对于任何$ n> n $，请考虑随机polytope $ k_n = {\ rm cons} \ {x_1，\ ldots，\ ldots，x_n \}根据$μ$。我们研究了一个问题，是否存在预期的$ k_n $的阈值。我们的方法基于$μ$的cramer变换$λ_μ^{\ ast} $。我们检查了$λ_μ^{\ ast} $的所有订单的存在，并在某些条件下确立了期望的急剧阈值$ {\ mathbb e} _ {μ^n} [μ（k_n）$ $ k_n $：IT接近$ 0 $ $ \ ln n \ ll \ ll \ ll \ ll \ ll \ ll \ ll \ ll \ ll。 e} _ {μ}（λ_μ^{\ ast}）$，接近$ 1 $，如果$ \ ln n n \ gg {\ mathbb e} _ {μ}（λ_μ^{\ ast}）$。主要条件是参数$β（μ）= {\ rm var} _ {μ}（λ_μ^{\ ast}）/（{\ Mathbb e} _ {μ} _ {μ}（λ_{μ}^{μ}^{\ ast}）
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.
