An Extended Dichotomy Theorem for Sequences of Pairs of Gaussian Measures

Abstract

A dichotomy for sequences of pairs of Gaussian measures is proved. This result is then used to give a simple proof of the famous equivalence/singularity dichotomy for Gaussian processes. The proof uses tightness arguments and can be directly applied to the theory of hypothesis testing to show that two sequences of simple hypotheses which specify Gaussian measures are either contiguous or entirely separable.