2025/10/14 by Armine Bagyan, Donald Richards, Bagyan, Armine +1
#math.ST #math.PR #stat.TH
paper · pdf · doi:10.48550/arxiv.2510.12928
We consider the modulation of data given by random vectors Xn ∈ ℝdn, n ∈ ℕ. For each Xn, one chooses an independent modulating random vector Ξn ∈ ℝdn and forms the projection Yn = Ξn'Xn. It is shown, under regularity conditions on Xn and Ξn, that Yn|Ξn converges weakly in probability to a normal distribution. More broadly, the conditional joint distribution of a family of projections constructed from random samples from Xn and Ξn is shown to converge weakly to a matrix normal distribution. We derive, via G. Pólya's characterization of the normal distribution, a necessary and sufficient condition on Yn for Ξn to be normally distributed. When Ξn 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 Yn|Ξn converges pointwise in certain pth means to a mixture of normal densities and the rate of convergence is quantified, resulting in uniform convergence. The cumulative distribution function of Yn|Ξn is shown to converge uniformly in those pth 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 Ξn include the multivariate t-, multivariate Laplace, and spherically symmetric stable distributions.