We consider an even probability distribution on the d-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given N independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension d and the number N of random vectors tend to infinity. In a similar way we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of k-faces and of Grassmann angles of index \(d-k\) when also k tends to infinity.

我们考虑d维欧几里得空间上的偶数概率分布，其特性是它为通过原点的任何超平面分配测度零。给定N个具有这种分布的独立随机向量，在它们不正跨越整个空间的条件下，这些向量的正包是一个随机多面体锥体（它与单位球体的交点是一个随机球面多面体）。Cover和Efron首先研究了它。我们考虑这些随机锥的预期面数，并描述了当维度d和数量N时的阈值现象的随机向量趋于无穷大。我们以类似的方式处理立体角，更一般地处理格拉斯曼角。当k也趋于无穷大时，我们进一步考虑k面的预期数量和指数\(dk\)的格拉斯曼角的数量。