Projections of the uniform distribution on the cube: a large deviation perspective
Tom 264 / 2022
Streszczenie
Let ${\mit\Theta} ^{(n)}$ 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 ${\mit\Theta} ^{(n)}$. The projected distribution is the random probability measure $\mu _{{\mit\Theta} ^{(n)}}$ on $\mathbb R$ given by \[ \mu _{{\mit\Theta} ^{(n)}}(A) := \frac 1 {2^n} \int _{[-1,1]^n} \mathbf {1} \{\langle u, {\mit\Theta} ^{(n)} \rangle \in A\} \,{\rm d} u \] for Borel subets $A$ of $\mathbb {R}$. It is well known that, with probability $1$, the sequence of random probability measures $\mu _{{\mit\Theta} ^{(n)}}$ converges weakly to the centered Gaussian distribution with variance $1/3$. We prove a large deviation principle for the sequence $\mu _{{\mit\Theta} ^{(n)}}$ 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 $$ \sqrt {1 - \|\alpha \|_2^2 } \frac {Z}{\sqrt 3} + \sum _{ k = 1}^\infty \alpha _k U_k, $$ where $Z$ is standard Gaussian independent of $U_1,U_2,\ldots $ which are i.i.d. ${\rm Unif} [-1,1]$, and $\alpha _1 \geq \alpha _2 \geq \cdots $ is a non-increasing sequence of non-negative reals with $\|\alpha \|_2 \lt 1$. We obtain a similar result for random projections of the uniform distribution on the discrete cube $\{-1,+1\}^n$.
