Feldman–Hájek theorem

In probability theory, the Feldman–Hájek theorem or Feldman–Hájek dichotomy is a fundamental result in the theory of Gaussian measures. It states that two Gaussian measures and on a locally convex space are either equivalent measures or else mutually singular:[1] there is no possibility of an intermediate situation in which, for example, has a density with respect to but not vice versa. In the special case that is a Hilbert space, it is possible to give an explicit description of the circumstances under which and are equivalent: writing and for the means of and , and and for their covariance operators, equivalence of and holds if and only if[2]

  • and have the same Cameron–Martin space ;
  • the difference in their means lies in this common Cameron–Martin space, i.e. ; and
  • the operator is a Hilbert–Schmidt operator on .

A simple consequence of the Feldman–Hájek theorem is that dilating a Gaussian measure on an infinite-dimensional Hilbert space (i.e. taking for some scale factor ) always yields two mutually singular Gaussian measures, except for the trivial dilation with , since is Hilbert–Schmidt only when .

References

  1. Bogachev, Vladimir I. (1998). Gaussian Measures. Mathematical Surveys and Monographs. 62. Providence, RI: American Mathematical Society. doi:10.1090/surv/062. ISBN 0-8218-1054-5. (See Theorem 2.7.2)
  2. Da Prato, Giuseppe; Zabczyk, Jerzy (2014). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. 152 (Second ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9781107295513. ISBN 978-1-107-05584-1. (See Theorem 2.25)
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