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Random cones in high dimensions I: Donoho-Tanner and Cover-Efron cones

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.