High-dimensional limit theorems for random vectors in ℓpn-balls

Abstract

In this paper, we prove a multivariate central limit theorem for ℓq-norms of highdimensional random vectors that are chosen uniformly at random in an ℓ n p -ball. As a consequence, we provide several applications on the intersections of ℓ n p -balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an ℓ n p -ball onto a line spanned by a random direction θ ∈ S n−1 . The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime 1 ≤ p < q this displays in speed and rate function deviations of the q-norm on an ℓ n p -ball obtained by Schechtman and Zinn, but we obtain explicit constants.2010 Mathematics Subject Classification. 52A22, 60D05, 60F05, 60F10.

High-dimensional limit theorems for random vectors in ℓpn-balls. II

“…These new findings for the moderate scaling therefore complement and refine both the new central limit theorems (Theorem A and Theorem B) as well the new large deviations principle (Theorem D). For a variety of applications of such results, despite the once presented below, we refer the reader to [13]. Before we present our results, let us explain the distributional set-up of this manuscript.…”

“…There is a central limit theorem due to Paouris, Pivovarov, and Zinn [18] for the volume of k-dimensional random projections of the n-dimensional cube when n → ∞, a result that had previously been obtained by Kabluchko, Litvak, and Zaporozhets [12] in the special case k = 1. Alonso-Gutiérrez, Prochno, and Thäle [1] proved a central limit theorem and Berry-Esseen bounds for the Euclidean norm of random orthogonal projections of points chosen uniformly at random from the unit ball of ℓ n p , as n → ∞, and Kabluchko, Prochno, and Thäle [13] obtained a multivariate central limit theorem for the q-norm of random vectors chosen uniformly at random in the unit p-ball of R n , which extended the corresponding 1-dimensional result obtained by Schmuckenschläger [22]. While the results in the previous paragraph describe central limit phenomena for several geometry related quantities, there is considerably less known about the large deviations behavior.…”

“…Before we present our results, let us explain the distributional set-up of this manuscript. As already mentioned, we consider a much more general class of distributions compared to [13] and [22]. Those have been introduced and studied by Barthe, Guédon, Mendelson, and Naor [6], and are closely related to the geometry of ℓ n pballs.…”

“…Those have been introduced and studied by Barthe, Guédon, Mendelson, and Naor [6], and are closely related to the geometry of ℓ n pballs. This class contains the uniform distribution considered in [1,11,13], the cone probability measure on the ℓ n p -unit ball B n p := {x ∈ R n : x p ≤ 1} as special cases, and many more (see below). As usual, x p = (|x 1 | p + .…”

“…If for each n, W n = W for some fixed Borel probability measure W on [0, ∞), then assumption (3) is clearly satisfied. In particular, taking W to be the Dirac measure at zero (recall (ii) above) or the exponential distribution with rate 1/p (recall (i) above), we recover the central limit theorem of Schmuckenschläger [22], see also Kabluchko, Prochno, and Thäle [13]. As another example, we fix a sequence of positive real numbers (a n ) n∈N such that a n / √ n → 0, as n → ∞, and let for each n ∈ N, W n = Γ(a n , b) be the gamma distribution with shape parameter a n and some fixed rate b ∈ (0, ∞).…”