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 $q$-norm of random vectors distributed on the ${\ell }_p^n$-ball ${\mathbb{B}}_p^n$ 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 Thäle 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 Schmuckenschläger, 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 to integrate these densities and derive concrete probability estimates.

28 pages, Updated Version: Reworked the proof of main result for ${\ell }_p^n$-balls in section 7 to be based on a result more appropriate to the given problem

References in corpus (3)

• A probabilistic approach to the geometry of the \ell_p^n-ball
• A Central Limit Theorem for Convex Sets
• Geometric sharp large deviations for random projections of ${\ell }_p^n$ spheres and balls