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

Samuel G. G. Johnston, Zakhar Kabluchko, Joscha Prochno

Published 2021-03-30Version 1

For $n\in\mathbb N$ let $\Theta^{(n)}$ be a random vector uniformly distributed on the unit sphere $\mathbb S^{n-1}$, and consider the associated random probability measure $\mu_{\Theta^{(n)}}$ given by setting \[ \mu_{\Theta^{(n)}}(A) := \mathbb{P} \left[ \langle U, \Theta^{(n)} \rangle \in A \right],\qquad U \sim \text{Unif}([-1,1]^n) \] for Borel subets $A$ of $\mathbb{R}$. It is known that 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 $\mu_{\Theta^{(n)}}$ with speed $n$ and explicit good rate rate function 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 projections of the uniform distribution on the discrete cube $\{-1,+1\}^n$.