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Large deviations. (English) Zbl 0949.60001

Fields Institute Monographs. 14. Providence, RI: American Mathematical Society (AMS). x, 143 p. (2000).
This is an introduction to the theory and the applications of large deviations, a branch of probability theory that describes the probability of rare events in terms of variational problems. Large deviations are an active research field and have many applications in statistics, operations research, ergodic theory, information theory and statistical mechanics, and more and more applications are being found from year to year. There are intimate connections to other branches of mathematics like variational calculus, convex analysis and set topology, and this makes this topic important, appealing and exciting. There are monographs on large deviations that contain much more, much more abstract and deeper results, but both generality and comprehension were not the aim of the author. Rather he wrote a user-friendly, concise and clear exposition of the spirit of this topic, avoiding a great deal of technicalities, paying with less wide applicability, but earning much greater accessibility. In fact, the book yields a quick understanding of the matter under restricted assumptions and also seems to be well suited for third-year students.
The book consists of ten chapters, the first five of which are lectures on the general theory, and the last five each explain a certain application in the research (mainly) of the nineties. In Chapters I and II, i.i.d. sequences with countable state space are considered, and basic results like Cramér’s and Sanov’s theorems are derived via explicit calculation. Chapter III puts the theory into a broader view and derives other basic results like the contraction principle, Varadhan’s lemma, some convexity assertions and a more general version of Cramér’s theorem. In Chapter IV the theory is extended to Markov chains via a change-of-measure technique, and Chapter V further extends it to moderately dependent sequences via the Gärtner-Ellis theorem.
In the five chapters on applications, all but the first one (statistical hypothesis testing) stem from research of the nineties the author was involved in: random walks in random environments, asymptotic correlations for the Cauchy problem with random potential, one-dimensional polymer chains, and interacting diffusions. Each chapter clearly explains what large deviations achieve for the respective model and gives a concise survey of the proofs. Several improvements in comparison to the original papers flew into the expositions.
Some 60 smaller and bigger exercises are placed at suited points in both parts, and comments on their solutions are provided in the Appendix. Lots of side-remarks shed light on the matter from some different points of view, and give hints to more specialized literature on particular topics.
Reviewer: W.König (Berlin)

MSC:

60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60F10 Large deviations
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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