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.
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© 1987 Springer-Verlag Berlin Heidelberg
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Jacod, J., Shiryaev, A.N. (1987). Hellinger Processes, Absolute Continuity and Singularity of Measures. In: Limit Theorems for Stochastic Processes. Grundlehren der mathematischen Wissenschaften, vol 288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02514-7_4
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DOI: https://doi.org/10.1007/978-3-662-02514-7_4
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