A Sanov-type theorem for empirical measures associated with the surface and cone measures on $\ell^{p}$ spheres

Steven Soojin Kim, Kavita Ramanan

Published 2015-09-17Version 1

We prove a large deviations principle (LDP) for the empirical measure of the coordinates of a random vector distributed according to the surface measure on a suitably scaled $\ell^p$ sphere in $\mathbb{R}^n$, as $n\rightarrow\infty$. This LDP is established for $p\in[1,\infty]$, with respect to the $q$-Wasserstein topology, for every $q < p$. We prove the result by first establishing an analogous LDP when the random vector is distributed according to the cone measure on the scaled $\ell^p$ sphere. In addition, we combine our LDP with the Gibbs conditioning principle to obtain an asymptotic probabilistic description of the geometry of an $\ell^p$ sphere under certain $\ell^q$ norm constraints, for $q<p$. These results are also of relevance for the study of the large deviations behavior of random projections of $\ell^p$ balls.