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 $\{ -1,+1\}^n$.
Categories: math.PR math.FA