  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.
