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, \textit{via} G. P\'olya's characterization of the normal distribution, a necessary and sufficient condition on $Y_n$ for $\Xi_n$ to be normally distributed. 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 $p$th means to a mixture of normal densities and the rate of convergence is quantified, resulting in uniform convergence. The cumulative distribution function of $Y_n|\Xi_n$ 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 $\Xi_n$ include the multivariate $t$-, multivariate Laplace, and spherically symmetric stable distributions.

翻译：我们考虑由随机向量$X_n \in \mathbb{R}^{d_n}$, $n \in \mathbb{N}$给出的数据调制问题。对于每个$X_n$，选择一个独立的调制随机向量$\Xi_n \in \mathbb{R}^{d_n}$并构造投影$Y_n = \Xi_n'X_n$。研究表明，在$X_n$和$\Xi_n$满足正则性条件下，$Y_n|\Xi_n$在概率意义下弱收敛于正态分布。更广泛地，由$X_n$和$\Xi_n$的随机样本构造的投影族之条件联合分布被证明弱收敛于矩阵正态分布。我们通过G. Pólya对正态分布的特征刻画，推导出$\Xi_n$服从正态分布时$Y_n$的充要条件。当$\Xi_n$具有球对称分布时，我们通过I. J. Schoenberg对希尔伯特空间上球对称特征函数的特征刻画，推导出$Y_n|\Xi_n$的概率密度函数在特定$p$阶均值意义下逐点收敛于正态密度混合分布，并量化了收敛速率，从而得到一致收敛性。$Y_n|\Xi_n$的累积分布函数被证明在相应$p$阶均值意义下一致收敛于同一混合分布的分布函数，并获得了Lipschitz性质。我们提供了满足研究结果的分布实例，包括随机半径超球面上的Bingham分布、随机体积超球面与超立方体上的均匀分布以及多元正态分布；此类$\Xi_n$的实例包括多元$t$-分布、多元拉普拉斯分布以及球对称稳定分布。