Hellinger Processes, Absolute Continuity and Singularity of Measures

Abstract

The question of absolute continuity or singularity (ACS) of two probability measures has been investigated a long time ago, both for its theoretical interest and for its applications to mathematical statistics. S. Kakutani in 1948 [125] was the first to solve the ACS problem in the case of two measures P and P′ having a (possibly infinite) product form: P = µ ₁ ⊗ µ ₂ ⊗ ... and P′ = µ′₁ ⊗ µ′₂ ⊗ ..., when µ _n ~ µ′ _n (µ _n and µ′ _n are equivalent) for all n; he proved a remarquable result, known as the “Kakutani alternative”, which says that either P ~ P, or P ⊥ P′ (P and P′ are mutually singular). Ten years later, Hajek [80] and Feldman [53] proved a similar alternative for Gaussian measures, and several authors gave effective criteria in terms of the covariance functions or spectral quantities, for the laws of two Gaussian processes.