{"ID":2855936,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.12928","arxiv_id":"2510.12928","title":"Random Modulation with Spherical Symmetry","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.","short_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 $Y_n|Ξ_n$...","url_abs":"https://arxiv.org/abs/2510.12928","url_pdf":"https://arxiv.org/pdf/2510.12928v1","authors":"[\"Armine Bagyan\",\"Donald Richards\"]","published":"2025-10-14T19:04:01Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\"]","methods":"[]","has_code":false}