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

TL;DR

This paper establishes a large deviation principle for the distribution of projections of high-dimensional uniform distributions on cubes, revealing the probability of deviations from the typical Gaussian limit in terms of an explicit rate function.

Contribution

It provides the first large deviation analysis for the projected uniform distribution on the cube, with an explicit rate function characterizing rare deviations.

Findings

Large deviation principle with speed n for projected distributions

Explicit rate function involving Gaussian and uniform variables

Extension to projections of the discrete cube ext{{-1,+1 extasciicircum n}}

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 \[ \mu_{\Theta^{(n)}}(A) := \frac 1 {2^n} \int_{[-1,1]^n} \mathbb 1\{\langle u, \Theta^{(n)} \rangle \in A\} du, \] 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 $I(\nu(\alpha)) := - \frac{1}{2} \log ( 1 - \|\alpha\|_2^2)$ whenever $\nu(\alpha)$ is the law of a random variable of the form \begin{align*} \sqrt{1 - \|\alpha\|_2^2 } \frac{Z}{\sqrt 3} + \sum_{ k = 1}^\infty \alpha_k U_k, \end{align*} where $Z$ is standard Gaussian independent of $U_1,U_2,\ldots$ which are i.i.d. $\text{Unif}[-1,1]$, and $\alpha_1 \geq \alpha_2 \geq \ldots $ is a non-increasing sequence of non-negative reals with $\|\alpha\|_2<1$. We obtain a similar result for random projections of the uniform distribution on the discrete cube $\{-1,+1\}^n$.