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A generalization of Laplace's method

Author: Chii Ruey Hwang
Journal: Proc. Amer. Math. Soc. 82 (1981), 446-451
MSC: Primary 60B05; Secondary 28C20
MathSciNet review: 612737
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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 }\vert\theta > 0\} $, with $ (d{P_\theta }/dQ)(x) = {C_\theta }\exp ( - H(x)/\theta )$, is considered, where

$\displaystyle H(x) = \frac{1} {2}\langle Fx,x\rangle - 2\langle Fm,x\rangle ),$

$ 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.

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Keywords: Characteristic function, covariance operator, Gaussian measure, fiber measure, Hilbert space, Laplace's method, weak convergence
Article copyright: © Copyright 1981 American Mathematical Society

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