高维随机锥I：Donoho-Tanner锥和Cover-Efron锥
Random cones in high dimensions I: Donoho-Tanner and Cover-Efron cones
摘要：
本文考虑了两种高维随机锥模型，并讨论了它们的分布。Donoho-Tanner随机锥$D_{n,d}$可以定义为$n$独立的$d$维高斯随机向量的正壳。Cover-Efron随机锥$C_{n,d}$本质上被定义为相同的正壳，条件是它不是整个空间。我们认为这些随机锥的各种组合和几何泛函的期望，并证明他们满足极限定理，$d$和$n$趋于无穷大的适当协调的方式。这包括，例如，大偏差原理和中心以及非中心极限定理的预期数量的$k$ -面和$k$ -th圆锥固有体积，因为$n$，$d$和可能还有$k$趋于无穷大同时.此外，我们确定精确的高维渐近行为的预期统计尺寸的随机锥模型，从而揭示另一个高维相变。作为应用，还讨论了由随机Gale图生成的高维多面体的$k$ -面数的极限定理.
英文摘要：
Two models of random cones in high dimensions are considered, together with their duals. The Donoho-Tanner random cone $D_{n,d}$ can be defined as the positive hull of $n$ independent $d$-dimensional Gaussian random vectors. The Cover-Efron random cone $C_{n,d}$ is essentially defined as the same positive hull, conditioned on the event that it is not the whole space. We consider expectations of various combinatorial and geometric functionals of these random cones and prove that they satisfy limit theorems, as $d$ and $n$ tend to infinity in a suitably coordinated way. This includes, for example, large deviation principles and central as well as non-central limit theorems for the expected number of $k$-faces and the $k$-th conic intrinsic volumes, as $n$, $d$ and possibly also $k$ tend to infinity simultaneously. Furthermore, we determine the precise high-dimensional asymptotic behaviour of the expected statistical dimension for both models of random cones, uncovering thereby another high-dimensional phase transition. As an application, limit theorems for the number of $k$-faces of high-dimensional polytopes generated by random Gale diagrams are discussed as well.
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