Sharp Asymptotics for $q$-Norms of Random Vectors in High-Dimensional $\ell_p^n$-Balls

Abstract

Sharp large deviation results of Bahadur & Ranga Rao-type are provided for the qnorm of random vectors distributed on the ℓ n p -ball B n p according to the cone probability measure or the uniform distribution for 1 ≤ q < p < ∞, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different ℓ n p -balls in the spirit of Schechtman and Schmuckenschläger, and for the length of the projection of an ℓ n p -ball onto a line with uniform random direction. The sharp large deviation results are proven by providing convenient probabilistic representations of the q-norms, employing local limit theorems to approximate their densities, and then using geometric results for asymptotic expansions of Laplace integrals of Adriani & Baldi and Liao & Ramanan to integrate over the densities and derive concrete probability estimates.

Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions

“…Moreover, an interesting connection between the study of large (and moderate) deviations for logconcave distributions and the famous Kannan-Lovász-Simonovits conjecture was established in [7]. Other than that a variety of large deviation results have been obtained in the last five years, among others, [24,41,42,43,46,50,51,52,57]. Beyond that, in [39] and the subsequent works [4,38], it has been demonstrated how ideas and methods from large deviation theory, such as the maximum entropy principle, its relation to Gibbs measures, and Gibbs conditioning, allow one to lift classical results for ℓ p -balls to more general symmetric Banach spaces (similar ideas have recently been used by Barthe and Wolff [12] studying Orlicz spaces).…”