Hellinger Processes, Absolute Continuity and Singularity of Measures
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  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  and Feldman  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.
KeywordsProbability Measure Point Process Strict Sense Local Uniqueness Absolute Continuity
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