Sharp asymptotics for q -norms of random vectors in high-dimensional -balls
Volume 8, Issue 2 (2021), pp. 239–274
Pub. online: 22 June 2021
Type: Research Article
Open Access
Open Access
Received
29 January 2021
29 January 2021
Revised
5 May 2021
5 May 2021
Accepted
22 May 2021
22 May 2021
Published
22 June 2021
22 June 2021
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.
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