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Laplace approximations for sums of independent random vectors. II: Degenerate maxima and manifolds of maxima


Bolthausen, E (1987). Laplace approximations for sums of independent random vectors. II: Degenerate maxima and manifolds of maxima. Probability Theory and Related Fields, 76(2):167-206.

Abstract

We consider expressions of the form Z n =E(exp nF(S n /n)) where S n is the sum of n i.i.d. random vectors with values in a Banach space and F is a smooth real valued function. By results of Donsker-Varadhan and Bahadur-Zabell one knows that lim (1/n) log Z n =sup x F(x)-h(x) where h is the so-called entropy function. In an earlier paper a more precise evaluation of Z n is given in the case where there was a unique point maximizing F-h and the curvature at the maximum was nonvanishing. The present paper treats the more delicate problem where these conditions fail to hold.

We consider expressions of the form Z n =E(exp nF(S n /n)) where S n is the sum of n i.i.d. random vectors with values in a Banach space and F is a smooth real valued function. By results of Donsker-Varadhan and Bahadur-Zabell one knows that lim (1/n) log Z n =sup x F(x)-h(x) where h is the so-called entropy function. In an earlier paper a more precise evaluation of Z n is given in the case where there was a unique point maximizing F-h and the curvature at the maximum was nonvanishing. The present paper treats the more delicate problem where these conditions fail to hold.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Laplace approximations; degenerate maxima; manifolds of maxima; entropy function
Language:English
Date:1987
Deposited On:20 Oct 2009 13:56
Last Modified:05 Apr 2016 13:29
Publisher:Springer
ISSN:0178-8051
Additional Information:The original publication is available at www.springerlink.com
Publisher DOI:https://doi.org/10.1007/BF00319983
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0608.60018

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