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

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 $\{-1,+1\}^n$.