2021/03/30 by Samuel G. G. Johnston, Johnston, Samuel G. G., Zakhar Kabluchko +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #52A23 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Primary 60F10 #Probability (math.PR) #Probability and Risk Models #Secondary 46B06 #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2103.16430
openalex publication_date 2021/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let Θ(n) be a random vector uniformly distributed on the unit sphere \mathbb Sn-1 in \mathbb 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 \mathbb R given by μΘ(n)(A) := \frac 1 2n ∫[-1,1]n \mathbb 1\⟨ 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 \mathbb R with speed n. The (good) rate function is explicitly given by I(ν(α)) := - (1)/(2) log ( 1 - ‖α‖22) whenever ν(α) is the law of a random variable of the form √(1 - ‖α‖22 ) (Z)/(√ 3) + ∑ k = 1^∞ αk Uk, 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.