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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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.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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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: | 10.1007/BF00319983 |

Related URLs: | http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0608.60018 |

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