CONTIGUITY under HIGH-DIMENSIONAL GAUSSIANITY with APPLICATIONS to COVARIANCE TESTING

Le Cam s third/contiguity lemma is a fundamental probabilistic tool to compute the limiting distribution of a given statistic Tn under a nonnull sequence of probability measures {Qn}, provided its limiting distribution under a null sequence {Pn} is available, and the log likelihood ratio {log(dQn/dPn)} has a distributional limit. Despite its wide-spread applications to low-dimensional statistical problems, the stringent requirement of Le Cam s third/contiguity lemma on the distributional limit of the log likelihood ratio makes it challenging, or even impossible to use in many modern highdimensional statistical problems.