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