Gaussian fluctuations for high-dimensional random projections of $\ell_{p}^{n}$-balls

Abstract

In this paper, we study high-dimensional random projections of ℓ n p -balls. More precisely, for any n ∈ N let En be a random subspace of dimension kn ∈ {1, . . . , n} and Xn be a random point in the unit ball of ℓ n p . Our work provides a description of the Gaussian fluctuations of the Euclidean norm PE n Xn 2 of random orthogonal projections of Xn onto En. In particular, under the condition that kn → ∞ it is shown that these random variables satisfy a central limit theorem, as the space dimension n tends to infinity. Moreover, if kn → ∞ fast enough, we provide a Berry-Esseen bound on the rate of convergence in the central limit theorem. At the end we provide a discussion of the large deviations counterpart to our central limit theorem.2010 Mathematics Subject Classification. Primary: 52A22 Secondary: 46B07, 60F05.

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

“…n / √ n both converge to zero in probability, as n → ∞. Moreover, the central limit theorem implies that S (1) n and S (2) n converge in distribution to non-degenerate Gaussian random variables. Hence, Slutsky's theorem implies that (S…”

“…Moreover, we denote by P En X n the orthogonal projection of X n onto E n . The quantity studied in [1,2] is the Euclidean norm of the projection of the random vector X n onto the random subspace E n , i.e., P En X n 2 . We first rephrase the central limit theorem [2, Theorem 1.1].…”

“…where the normalization constant c p is given by c p := 2p 1/p Γ(1 + 1 p ). Next, recall the definition of the constant M p (q) from (2). It can be used to express first-and second-order moments of p-generalized Gaussian random variables as follows.…”

“…Assumption (3) says that W n / √ n converges in distribution to 0, as n → ∞. Therefore, the multivariate central limit theorem applied to (S (1) n , S (2) n ) and the continuous mapping theorem yield…”

“…We start with the assertion on the sequence T n . First of all, we notice that by Assumption (4), the multivariate central limit theorem applied to (S (1) n , S (2) n ), and the continuous mapping theorem,…”