  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.
