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.

翻译：对称性是数学的基石之一，许多概率分布具有以其在群作用下的不变性为特征的对称性。因此，许多数学和统计方法依赖这种对称性成立，而当对称性被打破时，这些方法通常失效。本研究探讨了概率测度序列在何种条件下渐近地获得这种对称性或对群作用的不变性。鉴于高斯分布的多种对称性，本文有效提出了一种非参数类型的中心极限定理：即高维随机向量的Lipschitz函数将渐近地不变于某些紧致拓扑群的作用。该结果的应用包括：$\ell_p^n$球内均匀随机点的部分重对数律，以及经典参数统计检验与其随机化版本之间的渐近等价性（即使不变性假设不成立时）。