A conditional limit theorem for high-dimensional ℓᵖ-spheres

Abstract

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

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

“…where the function Ψ p : D F → R is such that, for some M, δ > 0, we have |Ψ p (x, y, z)| ≤ M (x, y, z) 2 2 whenever (x, y, z) 2 2 < δ. Combining this with the representation (14) for Z n q proves the claim. 4.2.…”

Yet another note on the arithmetic-geometric mean inequality

“…While large deviations are extensively studied in probability theory (see, e.g., [4,5] and the references cited therein), they have not been considered -contrary to central limit theorems -in geometric functional analysis until the very recent paper by Gantert, Kim, and Ramanan [6]. Already shortly after, this work has been extended and complemented in [3,9,10,12,13]. In contrast to the universality in central limit theorems, the probabilities of (large) deviations on the scale of laws of large numbers, are non-universal, thus being sensitive to the distribution of the random variables considered.…”

“…Note that (11) implies (13) β s * +1 p = p 1 − pt * and β p − s * + 1 p = βt * . Now, using (13) to exclude t * from Λ(s * , t * ), we obtain…”

Geometry of ℓ p n -Balls $$\ell _p^n\,\text{-Balls}$$ : Classical Results and Recent Developments

“…The body of research on large deviation principles in asymptotic geometric analysis, which we have just described above, is complemented by another paper of Kim and Ramanan [16], in which they proved an LDP for the empirical measure of an n 1/p multiple of a point drawn from an ℓ n p -sphere with respect to the cone or surface measure. The rate function identified is essentially the so-called relative entropy perturbed by some p-th moment penalty (see [16,Equation (3.4)]).…”

“…The body of research on large deviation principles in asymptotic geometric analysis, which we have just described above, is complemented by another paper of Kim and Ramanan [16], in which they proved an LDP for the empirical measure of an n 1/p multiple of a point drawn from an ℓ n p -sphere with respect to the cone or surface measure. The rate function identified is essentially the so-called relative entropy perturbed by some p-th moment penalty (see [16,Equation (3.4)]). While this result is again in the commutative setting of the ℓ n p -balls, Kabluchko, Prochno, and Thäle [13] recently studied principles of large deviations in the non-commutative framework of self-adjoint and classical Schatten p-classes.…”