标题： 随机调制与球面对称性 显示英文标题

标题： Random Modulation with Spherical Symmetry

摘要： 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。对于每个$X_n$，选择一个独立的调制随机向量$\Xi_n \in \mathbb{R}^{d_n}$并形成投影$Y_n = \Xi_n'X_n$。 在 $X_n$ 和 $\Xi_n$ 上的正则性条件下，证明了 $Y_n|\Xi_n$ 在概率上弱收敛于正态分布。 更广泛地说，从 $X_n$ 和 $\Xi_n$ 中抽取的随机样本构造的一族投影的条件联合分布被证明弱收敛于矩阵正态分布。 我们推导出，\textit{通过}G。 Pólya 关于正态分布的特征，对$Y_n$的必要且充分条件，使得$\Xi_n$服从正态分布。 当 $\Xi_n$具有球面对称分布时，我们通过 I. J. Schoenberg 关于希尔伯特空间上球面对称特征函数的特征，得出$Y_n|\Xi_n$的概率密度函数在某些$p$次均值下逐点收敛于正态密度的混合形式，收敛速度被量化，从而得到一致收敛。 $Y_n|\Xi_n$的累积分布函数被证明在这些$p$个均值下一致收敛于相同混合分布的分布函数，并得到了一个 Lipschitz 性质。 提供了满足我们结果的分布示例；这些包括超球面上随机半径的 Bingham 分布，超球面和超立方体上随机体积的均匀分布，以及多元正态分布；此类$\Xi_n$的示例包括多元$t$-、多元 Laplace 和球对称稳定分布。 显示英文摘要

摘要： 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.