Sharp Asymptotics for $q$-Norms of Random Vectors in High-Dimensional ${\mathrm{\ell }}_p^n$-Balls

Sharp large deviation results of Bahadur-Ranga Rao type are provided for the $q$-norm of random vectors distributed on the ${\mathrm{\ell }}_p^n$-ball ${\mathbb{B}}_p^n$ according to the cone probability measure or the uniform distribution for $1\le q<p<\mathrm{\infty }$, 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 ${\mathrm{\ell }}_p^n$-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.