Random Modulation with Spherical Symmetry
Armine Bagyan, Donald Richards
TL;DR
The paper analyzes high-dimensional linear random modulation $Y_n = \Xi_n' X_n$ with independent modulators $\Xi_n$, deriving conditional weak convergence results as dimension grows. Under regularity conditions on $X_n$ (thin-shell and zero-overlap), $Y_n|\Xi_n$ converges to a normal law, and the joint projection matrix converges to a matrix normal with i.i.d. $\mathcal{N}_1(0,\sigma^2)$ entries; when $\Xi_n$ is spherically symmetric, $f_{Y_n|\Xi_n}$ converges to a scale-mixture of normals with mixing variable $V$ governed by Schoenberg’s representation, yielding uniform $p$-mean convergence and Lipschitz control for the CDF. The authors characterize a necessary and sufficient condition for $\Xi_n$ to be normal via Polya’s theorem, and provide quantitative rates of convergence in terms of the Gram matrices $A_{n,k}$ and the regularity conditions. A broad set of examples for $X_n$ (Bingham on spheres, uniform on balls/cubes, and MVN) and $\Xi_n$ (multivariate $t$, Laplace, and stable mixtures) illustrate the applicability, including explicit rate bounds and uniform convergence results for densities and distribution functions. The results offer a rigorous justification for normal-like behavior in high-dimensional projections and yield practical tools for inference when projection directions are random and possibly spherically distributed.
Abstract
We consider the modulation of data given by random vectors $X_n \in \mathbb{R}^{d_n}$, $n \in \mathbb{N}$. For each $X_n$, one chooses an independent modulating random vector $Ξ_n \in \mathbb{R}^{d_n}$ and forms the projection $Y_n = Ξ_n'X_n$. It is shown, under regularity conditions on $X_n$ and $Ξ_n$, that $Y_n|Ξ_n$ converges weakly in probability to a normal distribution. More broadly, the conditional joint distribution of a family of projections constructed from random samples from $X_n$ and $Ξ_n$ is shown to converge weakly to a matrix normal distribution. We derive, \textit{via} G. Pólya's characterization of the normal distribution, a necessary and sufficient condition on $Y_n$ for $Ξ_n$ to be normally distributed. When $Ξ_n$ has a spherically symmetric distribution we deduce, through I. J. Schoenberg's characterization of the spherically symmetric characteristic functions on Hilbert spaces, that the probability density function of $Y_n|Ξ_n$ converges pointwise in certain $p$th means to a mixture of normal densities and the rate of convergence is quantified, resulting in uniform convergence. The cumulative distribution function of $Y_n|Ξ_n$ is shown to converge uniformly in those $p$th means to the distribution function of the same mixture, and a Lipschitz property is obtained. Examples of distributions satisfying our results are provided; these include Bingham distributions on hyperspheres of random radii, uniform distributions on hyperspheres and hypercubes of random volumes, and multivariate normal distributions; and examples of such $Ξ_n$ include the multivariate $t$-, multivariate Laplace, and spherically symmetric stable distributions.