Asymptotic symmetry and group invariance for randomization

TL;DR

This paper investigates conditions under which sequences of probability measures become asymptotically invariant to group actions, extending the concept of symmetry in probability distributions and providing a non-parametric central limit theorem for high-dimensional functions.

Contribution

It introduces a framework for understanding asymptotic symmetry in probability measures and applies it to high-dimensional Gaussian-related functions and statistical test equivalences.

Findings

Lipschitz functions of high-dimensional vectors become invariant under group actions asymptotically.

Partial law of the iterated logarithm for points in $ ext{l}_p^n$-balls.

Asymptotic equivalence between classical and randomized statistical tests.

Abstract

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.