  We consider two models of random cones together with their duals. Let
$Y_1,\dots,Y_n$ be independent and identically distributed random vectors in
$\mathbb R^d$ whose distribution satisfies some mild condition. The random
cones $G_{n,d}^A$ and $G_{n,d}^B$ are defined as the positive hulls
$\text{pos}\{Y_1-Y_2,\dots,Y_{n-1}-Y_n\}$, respectively
$\text{pos}\{Y_1-Y_2,\dots,Y_{n-1}-Y_n,Y_n\}$, conditioned on the event that
the respective positive hull is not equal to $\mathbb R^d$. We prove limit
theorems for various expected geometric functionals of these random cones, as
$n$ and $d$ tend to infinity in a coordinated way. This includes limit theorems
for the expected number of $k$-faces and the $k$-th conic quermassintegrals, as
$n$, $d$ and sometimes also $k$ tend to infinity simultaneously. Moreover, we
uncover a phase transition in high dimensions for the expected statistical
dimension for both models of random cones.
