论文标题
高尺寸的随机锥I：Donoho-Tanner和Cover-Efron锥
Random cones in high dimensions I: Donoho-Tanner and Cover-Efron cones
论文作者
论文摘要
考虑到高维的两种随机锥模型以及它们的双重模型。 Donoho-Tanner随机锥$ d_ {n，d} $可以定义为$ n $ n $独立$ d $ d $二维高斯随机向量的正壳。 cover-efron随机锥$ c_ {n，d} $实质上定义为相同的正船体，以事件为条件，这不是整个空间。我们考虑对这些随机锥体的各种组合和几何功能的期望，并证明它们满足限制定理，因为$ d $和$ n $倾向于以适当的协调方式无限。例如，其中包括大型偏差原理和中央和非中心限制定理，预期的$ k $ faces和$ k $ th的圆锥圆锥固有量，为$ n $，$ d $，可能同时同时无限。此外，我们确定了两种随机锥模型的预期统计维度的精确高维渐近行为，从而发现了另一种高维相变。作为一个应用程序，还讨论了由随机大风图生成的高维多面体的$ k $ face数量的限制定理。
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.
