We derive the limit null distribution of the class of Sobolev tests of uniformity on the hypersphere when the dimension and the sample size diverge to infinity at arbitrary rates. The limiting non-null behavior of these tests is obtained for a sequence of integrated von Mises-Fisher local alternatives. The asymptotic results are applied to test for high-dimensional rotational symmetry and spherical symmetry. Numerical experiments illustrate the derived behavior of the uniformity and spherically symmetry tests under the null and under local and fixed alternatives.
Ebner et al. (2025) studied this question.