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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 onthe Stiefel manifold of orthonormal $k_n$-frames in $\mathbb{R}^n$, in threedifferent 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 ofrandom vectors satisfies a certain "asymptotic thin shell condition", weestablish annealed large deviation principles (LDPs) for the correspondingsequence of random projections in the constant regime, and for the sequence ofempirical measures of the coordinates of the random projections in thesublinear 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 superquadraticinteraction potential, or the uniform measure on suitably scaled $\ell_p^n$balls, for $p \in [1,\infty)$, and generalized Orlicz balls defined via asuperquadratic function. Our results complement the central limit theorem forconvex sets and related results which are known to hold under a "thin shell"condition. These results also substantially extend existing large deviationresults for random projections, which are first, restricted to the setting ofmeasures 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 orthe projection of $X^{(n)}$ onto a random one-dimensional subspace. Randomprojections of high-dimensional random vectors are of interest in a range offields including asymptotic convex geometry and high-dimensional statistics.

An asymptotic thin shell condition and large deviations for random multidimensional projections