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

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\"{a}le 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\"{a}ger, 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.