论文标题
随机分组的渐近对称性和群体不变性
Asymptotic symmetry and group invariance for randomization
论文作者
论文摘要
对称性是数学大部分的基石，许多概率分布都具有其对称性的对称性，其特征是它们的不变性与集体行动集合。因此，许多数学和统计方法都依赖于这种对称性保持，如果对称性破裂，则表面上是失败的。这项工作认为在什么条件下，一系列概率措施渐近地将这种对称性或不变性带入了集体行动的集合。考虑到高斯分布的许多对称性，这项工作有效地提出了一种非参数类型的中心极限定理。也就是说，高维随机矢量的Lipschitz函数将渐近地与某些紧凑型拓扑组的作用渐变。此应用包括迭代对数的部分定律，用于$ \ ell_p^n $ ball中均匀的随机点，即使经典的参数统计测试与它们的随机化对应物之间的渐近等效性，即使不变性假设违反了不变性假设。
Symmetry is a cornerstone of much of mathematics, and many probability distributions possess symmetries characterized by their invariance to a collection of group actions. Thus, many mathematical and statistical methods rely on such symmetry holding and ostensibly fail if symmetry is broken. This work considers under what conditions a sequence of probability measures asymptotically gains such symmetry or invariance to a collection of group actions. Considering the many symmetries of the Gaussian distribution, this work effectively proposes a non-parametric type of central limit theorem. That is, a Lipschitz function of a high dimensional random vector will be asymptotically invariant to the actions of certain compact topological groups. Applications of this include a partial law of the iterated logarithm for uniformly random points in an $\ell_p^n$-ball and an asymptotic equivalence between classical parametric statistical tests and their randomization counterparts even when invariance assumptions are violated.
