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

For $ninmathbb N$ 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 $I( u(alpha)) := - frac{1}{2} log ( 1 - ||alpha||_2^2)$ whenever $ u(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 ${-1,+1}^n$.