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

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

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