= Cameron-Martin theorem for a Gaussian measure
{c}
For $\mu=\mathcal N(0,\Sigma)$ with injective <covariance operator of a Gaussian measure>, its translated law $\mu_h=\mathcal N(h,\Sigma)$ is an <equivalent probability measure> to $\mu$ if and only if $h\in E_\mu$, the <Cameron-Martin space of a Gaussian measure>. For such $h$,
$$
\frac{d\mu_h}{d\mu}(x)=\exp\left(\ell_h(x)-\tfrac12\|h\|_{E_\mu}^2\right),\qquad
\ell_h(x)=\sum_j\frac{h_jx_j}{\lambda_j}.
$$
The series has <mean-square convergence> and converges <almost surely> under $\mu$, with <normal distribution> $\mathcal N(0,\|h\|_{E_\mu}^2)$. Its exponential density has <expected value> one by the <moment-generating function of a normal distribution>. Finite-dimensional projections give the formula by ratios of <multivariate normal densities>; convergence of these likelihood ratios gives the infinite-dimensional result. The expression $\langle h,x\rangle_{E_\mu}$ is only formal when the sample is outside $E_\mu$. For $h\notin E_\mu$, translation gives <mutually singular measures>.
Back to article page
