Feldman–Hájek theorem

Short description: Theory in probability theory

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: 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

• [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]