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**Large deviations.
Rev. ed.**
*(English)*
Zbl 0705.60029

Pure and Applied Mathematics, 137. Boston, MA etc.: Academic Press, Inc. xiv, 307 p. $ 34.95 (1989).

[The original edition by the second author appeared under the title “An introduction to the theory of large deviations” (1984; Zbl 0552.60022).]

This is not a book that will make you fall in love with Large Deviation Theory (LDT), but for those already acquainted with the theory, it could help the relationship mature. The book provides a sound base for LDT and answers questions and clears up technical problems found in articles previously written on the subject. Have you ever tried to figure out the precise topology used by M. D. Donsker and S. R. S. Varadhan in their article “Asymptotic evaluation of certain Markov process expectations for large time. IV”, Commun. Pure Appl. Math. 36, 183-212 (1983; Zbl 0512.60068)? The answer is on page 178 of this book. But by clarifying technicalities, the authors lose vision of the ideas and concepts that produced LDT. The historical connections between LDT and statistical mechanics, for instance, are neglected. The presentation separates it from the natural context in which the theory can be applied.

One example of this is the treatment given to the “Wiener sausage”, one of the motivations in the development of the Donsker Varadhan theory. You should read Barry Simon’s nice comment about it’s physical relevance, despite the fact that these authors refer to this as “a rather strange calculation” (see page 146). LDT is presented as a mathematical subject beyond all real relevance. The book only enlightens you on the “abstract spirit of large deviations”.

Chapter I presents two classical examples: the theorems of Cramér and Schilder. Both are used in order to introduce the basic strategies, the cornerstones for coping with large deviation problems:

exponential Chebyshev inequality; convex analysis; modification of probability measures; weak and strong LD principle; exponential tightness.

The power of Schilder’s theorem is demonstrated by Strassen’s law of iterated logarithm and in application for finding Wentzel Freidlin estimates. Chapter II is presented in Bourbaki-style and contains the general basis for the above mentioned techniques.

Chapter III is devoted to Sanov’s theorem and Cramér’s theorem for Banach space variables. Section 1 outlines a method for calculating the lower bound estimate which differs from the “modification strategy”. All in all, it is a good presentation of the Landford idea of using subadditive functions. Chapter IV is devoted to the work of Donsker and Varadhan. New techniques have been applied to prove their results. Landford’s ideas are also applicable here in order to get the lower bound estimates for Markov processes. Process large deviation results are proved by using projective limits of large deviation principles. This concept paves the way for a nice, short presentation of level III large deviations.

Chapter V introduces non-uniform LD results. The various processes presented here are not exclusively Markov processes and show a different ergodic behavior from those discussed in Chapter IV. And there are two highlights in this chapter: The first is the section on ergodic theory which proves the ergodic decomposition theorem and the second is the presentation of T. Chiyonobu and S. Kusuoka’s [Probab. Theory Relat. Fields 78, No.4, 627-649 (1988; Zbl 0634.60025)] LDT results for hypermixing processes. The hypermixing property was formulated and used while trying to construct quantum field theory. It is related to Nelson’s idea about hypercontractive semigroups. The connection between the two notions is explained first in section 5.5 for the \(\epsilon\)-Markov case and then again in section 6.1.

Chapter VI presents the theory of hypercontractive processes. Here you will find the interesting material, the reward for having read so far. One could liken the experience to that of climbing a mountain. After struggling with difficult technical demands, you then get to enjoy a grandiose view over a crystal landscape, where you can perceive traces of life way off in the distance.

This is not a book that will make you fall in love with Large Deviation Theory (LDT), but for those already acquainted with the theory, it could help the relationship mature. The book provides a sound base for LDT and answers questions and clears up technical problems found in articles previously written on the subject. Have you ever tried to figure out the precise topology used by M. D. Donsker and S. R. S. Varadhan in their article “Asymptotic evaluation of certain Markov process expectations for large time. IV”, Commun. Pure Appl. Math. 36, 183-212 (1983; Zbl 0512.60068)? The answer is on page 178 of this book. But by clarifying technicalities, the authors lose vision of the ideas and concepts that produced LDT. The historical connections between LDT and statistical mechanics, for instance, are neglected. The presentation separates it from the natural context in which the theory can be applied.

One example of this is the treatment given to the “Wiener sausage”, one of the motivations in the development of the Donsker Varadhan theory. You should read Barry Simon’s nice comment about it’s physical relevance, despite the fact that these authors refer to this as “a rather strange calculation” (see page 146). LDT is presented as a mathematical subject beyond all real relevance. The book only enlightens you on the “abstract spirit of large deviations”.

Chapter I presents two classical examples: the theorems of Cramér and Schilder. Both are used in order to introduce the basic strategies, the cornerstones for coping with large deviation problems:

exponential Chebyshev inequality; convex analysis; modification of probability measures; weak and strong LD principle; exponential tightness.

The power of Schilder’s theorem is demonstrated by Strassen’s law of iterated logarithm and in application for finding Wentzel Freidlin estimates. Chapter II is presented in Bourbaki-style and contains the general basis for the above mentioned techniques.

Chapter III is devoted to Sanov’s theorem and Cramér’s theorem for Banach space variables. Section 1 outlines a method for calculating the lower bound estimate which differs from the “modification strategy”. All in all, it is a good presentation of the Landford idea of using subadditive functions. Chapter IV is devoted to the work of Donsker and Varadhan. New techniques have been applied to prove their results. Landford’s ideas are also applicable here in order to get the lower bound estimates for Markov processes. Process large deviation results are proved by using projective limits of large deviation principles. This concept paves the way for a nice, short presentation of level III large deviations.

Chapter V introduces non-uniform LD results. The various processes presented here are not exclusively Markov processes and show a different ergodic behavior from those discussed in Chapter IV. And there are two highlights in this chapter: The first is the section on ergodic theory which proves the ergodic decomposition theorem and the second is the presentation of T. Chiyonobu and S. Kusuoka’s [Probab. Theory Relat. Fields 78, No.4, 627-649 (1988; Zbl 0634.60025)] LDT results for hypermixing processes. The hypermixing property was formulated and used while trying to construct quantum field theory. It is related to Nelson’s idea about hypercontractive semigroups. The connection between the two notions is explained first in section 5.5 for the \(\epsilon\)-Markov case and then again in section 6.1.

Chapter VI presents the theory of hypercontractive processes. Here you will find the interesting material, the reward for having read so far. One could liken the experience to that of climbing a mountain. After struggling with difficult technical demands, you then get to enjoy a grandiose view over a crystal landscape, where you can perceive traces of life way off in the distance.

Reviewer: U.Mansmann

### MSC:

60F10 | Large deviations |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60F15 | Strong limit theorems |