Large deviations for uniform projections of $p$-radial distributions on ${\mathrm{\ell }}_p^n$-balls

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 ${\mathrm{\ell }}_p^n$-ball ${\mathbb{B}}_p^n$ with certain $p$-radial distributions, $p\in [1,\mathrm{\infty })$. The distribution of this product geometrically corresponds to the projection of the $p$-radial distribution on ${\mathbb{B}}_p^n$ 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.