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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
DOI: https://doi.org/10.1090/S0002-9939-1981-0612737-8
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.

References [Enhancements On Off] (What's this?)

  • [1] N. Dunford and J. T. Schwartz, Linear operators, vol. II, Interscience, New York, 1963. MR 0188745 (32:6181)
  • [2] U. Grenander, Probabilities on algebraic structures, Wiley, New York, 1963. MR 0259969 (41:4598)
  • [3] -, Lectures in pattern theory, vol. III (to appear).
  • [4] -, Solve the second limit problem in metric pattern theory, Report on Pattern Analysis No. 83, Div. of Appl. Math., Brown University, Providence, R. I., 1979.
  • [5] C. R. Hwang, Frozen patterns and minimal energy states, Ph. D. thesis, Brown University, Providence, R. I., 1978.
  • [6] -, Laplace's method revisited: Weak convergence of probability measures, Ann. Probability 8 (1980).
  • [7] A. I. Khinchin, Mathematical foundation of statistical mechanics, Dover, New York, 1957. MR 0029808 (10:666c)
  • [8] P. Krée and A. Tortrat, Désintégration d'un loi gaussienne $ \mu $ dans une somme vectorielle, C. R. Acad. Sci. Paris Sér. A 277 (1973), 695-697. MR 0360994 (50:13441)
  • [9] B. S. Rajput, The support of Gaussian measures on Banach spaces, Theor. Probability Appl. 17 (1972), 728-734. MR 0324737 (48:3086)
  • [10] F. Riesz and B. Nagy, Functional analysis, McGraw-Hill, New York, 1955.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0612737-8
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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