Projections of the uniform distribution on the cube -- a large deviation perspective

Abstract

Let ${\Theta }^{(n)}$ be a random vector uniformly distributed on the unit sphere ${\mathbb{S}}^{n-1}$ in ${\mathbb{R}}^n$. Consider the projection of the uniform distribution on the cube $[-1,1{]}^n$ to the line spanned by ${\Theta }^{(n)}$. The projected distribution is the random probability measure ${\mu }_{{\Theta }^{(n)}}$ on $\mathbb{R}$ given by \[ μ_{Θ^{(n)}}(A) := \frac 1 {2^n} \int_{[-1,1]^n} \mathbb 1\{\langle u, Θ^{(n)} \rangle \in A\} du, \] for Borel subets $A$ of $\mathbb{R}$. It is well known that, with probability $1$, the sequence of random probability measures ${\mu }_{{\Theta }^{(n)}}$ converges weakly to the centered Gaussian distribution with variance $1/3$. We prove a large deviation principle for the sequence ${\mu }_{{\Theta }^{(n)}}$ on the space of probability measures on $\mathbb{R}$ with speed $n$. The (good) rate function is explicitly given by $I(\nu (\alpha )):=-\frac{1}{2}\log (1-\Vert \alpha {\Vert }_2^2)$ whenever $\nu (\alpha )$ is the law of a random variable of the form \begin{align*} \sqrt{1 - \|α\|_2^2 } \frac{Z}{\sqrt 3} + \sum_{ k = 1}^\infty α_k U_k, \end{align*} where $Z$ is standard Gaussian independent of $U_1,U_2,\dots$ which are i.i.d. $\text{Unif}[-1,1]$, and ${\alpha }_1\ge {\alpha }_2\ge \dots$ is a non-increasing sequence of non-negative reals with $\Vert \alpha {\Vert }_2<1$. We obtain a similar result for random projections of the uniform distribution on the discrete cube $\{-1,+1{\}}^n$.

12 pages