Hellinger Processes, Absolute Continuity and Singularity of Measures

  • Jean Jacod
  • Albert N. Shiryaev
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 288)


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 1µ 2 ⊗ ... and P′ = µ1µ2 ⊗ ..., 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 PP′ (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.


Probability Measure Point Process Strict Sense Local Uniqueness Absolute Continuity 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Jean Jacod
    • 1
  • Albert N. Shiryaev
    • 2
  1. 1.Laboratoire de ProbabilitésParis 05France
  2. 2.Steklov Mathematical InstituteMoscowUSSR

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