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

Tom Kaufmann

Published 2021-02-26Version 1

Sharp large deviation results of Bahadur & Ranga Rao-type are provided for the $q$-norm of random vectors distributed on the $\ell_p^n$-ball $\mathbb{B}^n_p$ according to the cone probability measure or the uniform distribution for $1 \le q<p < \infty$, thereby furthering previous large deviation results by Kabluchko, Prochno and Th\"ale in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different $\ell_p^n$-balls in the spirit of Schechtman and Schmuckenschl\"ager, and for the length of the projection of an $\ell_p^n$-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.