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On Cramér's Theory in Infinite Dimensions
Raphaël Cerf, Université Paris Sud, Orsay, France
A publication of the Société Mathématique de France.
Panoramas et Synthèses
2007; 159 pp; softcover
Number: 23
ISBN-10: 2-85629-235-6
ISBN-13: 978-2-85629-235-8
List Price: US$53
Member Price: US$42.40
Order Code: PASY/23
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This text is a self-contained account of Cramér's theory in infinite dimensions. The point of view is slightly different from the classical texts of Azencott, Bahadur and Zabell, Dembo and Zeitouni, and Deuschel and Stroock. The authors have been trying to understand the relevance of the topological hypotheses necessary to carry out the core of the theory. They have also drawn some inspiration from the analogy between the large deviation proofs in statistical mechanics and for i.i.d. random variables.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicans interested in probability and analysis.

Table of Contents

  • Introduction
  • Large deviation theory
  • Topological vector spaces
  • The model
  • The weak large deviation principle
  • The measurability hypotheses
  • Subadditivity
  • Proof of Theorem 5.2
  • Convex regularity
  • Enhanced upper bound
  • The Cramér transform \(I(\mu, A)\) as a function of \(\mu\)
  • The Cramér transform and the Log-Laplace
  • \(I=\Lambda^\ast\): the discrete case
  • \(I=\Lambda^\ast\): the smooth case
  • \(I=\Lambda^\ast\): the finite dimensional case
  • \(I=\Lambda^\ast\): the infinite dimensions
  • Exponential tightness
  • Cramér's theorem in \(\mathbb R\)
  • Cramér's theorem in \(\mathbb R^d\)
  • Cramér's theorem in the weak topology
  • Cramér's theorem in a Banach space
  • Gaussian measures
  • Sanov's theorem: autonomous derivation
  • Cramér's theorem implies Sanov's theorem
  • Sanov's theorem implies the compact Cramér theorem
  • Mosco convergence
  • A. Lusin's theorem
  • B. The mean of a probability measure
  • C. Ky Fan's proof of the minimax theorem
  • Index
  • Bibliography
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