A generalization of Laplace’s method

Abstract:

Let $Q$ be Gaussian with mean 0 and covariance $B$ in a separable Hilbert space. Analogous to Laplace’s method, the weak limit (as $\theta \downarrow 0$) $P$ of $\{ {P_\theta }|\theta > 0\}$, with $(d{P_\theta }/dQ)(x) = {C_\theta }\exp ( - H(x)/\theta )$, is considered, where \[ H(x) = \frac {1} {2}\langle Fx,x\rangle - 2\langle Fm,x\rangle ),\] $F$ is s.a. nonnegative definite and bounded. If $m \in \Re ({B^{1/2}})$, then $P$ is Gaussian with mean $m - {B^{1/2}}\pi {B^{ - 1/2}}m$ and covariance ${B^{1/2}}\pi {B^{1/2}}$, where $\pi$ is the projection onto $\mathfrak {N}({B^{1/2}}F{B^{1/2}})$. Moreover $P$ is the fiber measure of $Q$ on $m + \mathfrak {N}(F)$. Under stronger conditions, $P$ is induced by an affine transformation.