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 \(\Xi _n \in \mathbb {R}^{d_n}\) and forms the projection \(Y_n = \Xi _n'X_n\) . It is shown, under regularity conditions on \(X_n\) and \(\Xi _n\) , that \(Y_n|\Xi _n\) 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 \(\Xi _n\) 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 \(Y_n\) for \(\Xi _n\) to be normally distributed, and we show that our results motivate generalizations of Pólya’s theorem. When \(\Xi _n\) 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 \(Y_n|\Xi _n\) converges pointwise in certain pth means to a mixture of normal densities and a rate of convergence is quantified, resulting in uniform convergence. The cumulative distribution function of \(Y_n|\Xi _n\) 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 for \(X_n\) that satisfy our results include the Bingham distributions on hyperspheres of random radii, uniform distributions on hyperspheres and hypercubes of random volumes, and multivariate normal distributions; and examples of such \(\Xi _n\) include the multivariate t-, multivariate Laplace, and spherically symmetric stable distributions.
