Large deviations for uniform projections of $p$-radial distributions on $\ell_p^n$-balls

TL;DR

This paper establishes large deviation principles for the distribution of projections of $p$-radial distributions on $ ext{ell}_p^n$-balls onto random subspaces, combining geometric probability and large deviations theory.

Contribution

It introduces a novel large deviation framework for the projections of $p$-radial distributions on $ ext{ell}_p^n$-balls, linking geometric projections with probabilistic large deviations.

Findings

Derived large deviation principles for projections onto random subspaces.

Connected geometric projections with probabilistic large deviations.

Provided a new perspective on the behavior of $p$-radial distributions under projection.

Abstract

We consider products of uniform random variables from the Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^n$, $k \le n$, and random vectors from the $n$-dimensional $\ell_p^n$-ball $\mathbb{B}_p^n$ with certain $p$-radial distributions, $p\in[1,\infty)$. The distribution of this product geometrically corresponds to the projection of the $p$-radial distribution on $\mathbb{B}^n_p$ onto a random $k$-dimensional subspace. We derive large deviation principles (LDPs) on the space of probability measures on $\mathbb{R}^k$ for sequences of such projections.