Projections of the uniform distribution on the unit cube - a large deviations perspective

Samuel G. G. Johnston, Zakhar Kabluchko, Joscha Prochno

Research output: Contribution to journalArticlepeer-review

Abstract

Let Θ(n) be a random vector uniformly distributed on the unit sphere Sn−1 in Rn. Consider the projection of the uniform distribution on the cube [−1,1]n to the line spanned by Θ(n). The projected distribution is the random probability measure μΘ(n) on R given by
μΘ(n)(A):=12n∫[−1,1]n1{⟨u,Θ(n)⟩∈A}du
for Borel subets A of R. It is well known that, with probability 1, the sequence of random probability measures μΘ(n) converges weakly to the centered Gaussian distribution with variance 1/3. We prove a large deviation principle for the sequence μΘ(n) on the space of probability measures on R with speed n. The (good) rate function is explicitly given by I(ν(α)):=−12log(1−∥α∥22) whenever ν(α) is the law of a random variable of the form
1−∥α∥22−−−−−−−√Z3√+∑k=1∞αkUk,
where Z is standard Gaussian independent of U1,U2,… which are i.i.d. Unif[−1,1], and α1≥α2≥⋯ is a non-increasing sequence of non-negative reals with ∥α∥2<1. We obtain a similar result for random projections of the uniform distribution on the discrete cube {−1,+1}n.
Original languageEnglish
Pages (from-to)103-119
Number of pages17
JournalStudia Mathematica
Volume264
Early online date17 Dec 2021
DOIs
Publication statusE-pub ahead of print - 17 Dec 2021

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