  We consider products of uniform random variables from the Stiefel manifold of
orthonormal $k$-frames in $\mathbb{R}^n$, $k \le n$, and random vectors from
the $n$-dimensional $\ell_p^n$-ball $\mathbb{B}_p^n$ with certain $p$-radial
distributions, $p\in[1,\infty)$. The distribution of this product geometrically
corresponds to the projection of the $p$-radial distribution on
$\mathbb{B}^n_p$ onto a random $k$-dimensional subspace. We derive large
deviation principles (LDPs) on the space of probability measures on
$\mathbb{R}^k$ for sequences of such projections.
