  Consider the projection of an $n$-dimensional random vector onto a random
$k_n$-dimensional basis, $k_n \leq n$, drawn uniformly from the Haar measure on
the Stiefel manifold of orthonormal $k_n$-frames in $\mathbb{R}^n$, in three
different asymptotic regimes as $n \rightarrow \infty$: "constant" ($k_n=k$),
"sublinear" ($k_n \rightarrow \infty$ but $k_n/n \rightarrow 0$) and "linear"
$k_n/n \rightarrow \lambda$ with $0 < \lambda \le 1$). When the sequence of
random vectors satisfies a certain "asymptotic thin shell condition", we
establish annealed large deviation principles (LDPs) for the corresponding
sequence of random projections in the constant regime, and for the sequence of
empirical measures of the coordinates of the random projections in the
sublinear and linear regimes. We also establish LDPs for certain scaled
$\ell_q$ norms of the random projections in these different regimes. Moreover,
we verify our assumptions for various sequences of random vectors of interest,
including those distributed according to Gibbs measures with superquadratic
interaction potential, or the uniform measure on suitably scaled $\ell_p^n$
balls, for $p \in [1,\infty)$, and generalized Orlicz balls defined via a
superquadratic function. Our results complement the central limit theorem for
convex sets and related results which are known to hold under a "thin shell"
condition. These results also substantially extend existing large deviation
results for random projections, which are first, restricted to the setting of
measures on $\ell_p^n$ balls, and secondly, limited to univariate LDPs (i.e.,
in $\mathbb{R}$) involving either the norm of a $k_n$-dimensional projection or
the projection of $X^{(n)}$ onto a random one-dimensional subspace. Random
projections of high-dimensional random vectors are of interest in a range of
fields including asymptotic convex geometry and high-dimensional statistics.
