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arXiv · 2211.00144

Asymptotic symmetry and group invariance for randomization

Adam B Kashlak
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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.

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Adam B Kashlak. 2023-10-20. Asymptotic symmetry and group invariance for randomization. https://arxiv.org/abs/2211.00144

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