Random Modulation with Spherical Symmetry
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912647959543808 |
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| author | Bagyan, Armine Richards, Donald |
| author_facet | Bagyan, Armine Richards, Donald |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12928 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random Modulation with Spherical Symmetry Bagyan, Armine Richards, Donald Statistics Theory Probability 60F05, 62E20 (Primary) 62G20, 62H10 (Secondary) 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. |
| title | Random Modulation with Spherical Symmetry |
| topic | Statistics Theory Probability 60F05, 62E20 (Primary) 62G20, 62H10 (Secondary) |
| url | https://arxiv.org/abs/2510.12928 |