Sharp asymptotics for q-norms of random vectors in high-dimensional ℓpn-balls

Abstract

Sharp large deviation results of Bahadur-Ranga Rao type are provided for the q-norm of random vectors distributed on the l(p)(n)-ball B-p(n) according to the cone probability measure or the uniform distribution for 1 <= q < p < infinity, thereby furthering previous large deviation results by Kabluchko, Prochno and Thale in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different l(p)(n)-balls in the spirit of Schechtman and Schmuckenschlager, and for the length of the projection of an l(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.