Feldman–Hájek theorem

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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 [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \nu }[/math] on a locally convex space [math]\displaystyle{ X }[/math] are either equivalent measures or else mutually singular:[1] there is no possibility of an intermediate situation in which, for example, [math]\displaystyle{ \mu }[/math] has a density with respect to [math]\displaystyle{ \nu }[/math] but not vice versa. In the special case that [math]\displaystyle{ X }[/math] is a Hilbert space, it is possible to give an explicit description of the circumstances under which [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \nu }[/math] are equivalent: writing [math]\displaystyle{ m_{\mu} }[/math] and [math]\displaystyle{ m_{\nu} }[/math] for the means of [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \nu }[/math], and [math]\displaystyle{ C_\mu }[/math] and [math]\displaystyle{ C_\nu }[/math] for their covariance operators, equivalence of [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \nu }[/math] holds if and only if[2]

  • [math]\displaystyle{ \mu }[/math] and [math]\displaystyle{ \nu }[/math] have the same Cameron–Martin space [math]\displaystyle{ H = C_{\mu}^{1/2}(X) = C_{\nu}^{1/2}(X) }[/math];
  • the difference in their means lies in this common Cameron–Martin space, i.e. [math]\displaystyle{ m_{\mu} - m_{\nu} \in H }[/math]; and
  • the operator [math]\displaystyle{ (C_{\mu}^{-1/2} C_{\nu}^{1/2}) (C_{\mu}^{-1/2} C_{\nu}^{1/2})^{\ast} - I }[/math] is a Hilbert–Schmidt operator on [math]\displaystyle{ \bar{H} }[/math].

A simple consequence of the Feldman–Hájek theorem is that dilating a Gaussian measure on an infinite-dimensional Hilbert space [math]\displaystyle{ X }[/math] (i.e. taking [math]\displaystyle{ C_{\nu} = s C_{\mu} }[/math] for some scale factor [math]\displaystyle{ s \geq 0 }[/math]) always yields two mutually singular Gaussian measures, except for the trivial dilation with [math]\displaystyle{ s = 1 }[/math], since [math]\displaystyle{ (s^{2} - 1) I }[/math] is Hilbert–Schmidt only when [math]\displaystyle{ s = 1 }[/math].

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)