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Large deviations. (English) Zbl 1146.60003
This paper is based on the Wald Lectures diven at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations. The author has considered following sections of the theory:
1. Large deviations for sums;
2. Rate functions, duality and generating functions;
3. Markov processes;
4. Small random perturbations. The exit problem;
5. Gibbs measures and statistical mechanics;
6. Interacting particle systems;
7. Superexponential estimates;
8. Hydrodynamical limits;
9. Large deviations in hydrodynamical limits;
10. Large deviations for random walks in random environments;
11. Homogenization of Hamilton-Jacobi-Bellman equations;
12. History and references.
Remark: From the revierwer’s point of view, a survey on the modern theory of large deviations should also include 1) the classical paper by A. A. Borovkov [Th. Probab. Appl., 12, 575–595 (1967; Zbl 0178.20004)], which together with the papers by I. N. Sanov [Ref. 18, Mat. Sb., N. Ser. 42(84), 11–44 (1957; Zbl 0078.31202)] and by S. R. S. Varadhan [Ref. 20, Commun. Pure Appl. Math. 20, 659–685 (1967; Zbl 0278.60051)] lays in the basis of modern theory of large deviations; 2) the paper by A. A. Puhalskii [Theory Probab. Appl. 38, 490–497 (1993; Zbl 0807.60036)], in which necessary and sufficient conditions for large deviation principle are suggested.

MSC:
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60F10 Large deviations
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References:
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