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**Large deviations techniques and applications.
2nd ed.**
*(English)*
Zbl 0896.60013

Applications of Mathematics. 38. New York, NY: Springer. xvi, 396 p. (1998).

The book provides a very pleasant introduction to the theory of large deviations. It can serve as the basis for an undergraduate as well as for a graduate course on large deviations. Large deviation techniques are illustrated with many examples.

Chapter 1 is devoted to the introduction and definition of a large deviation principle (LDP). Chapter 2 treats LDP’s on finite-dimensional spaces; the main example is the empirical mean of a sequence of random variables taking values in \(\mathbb{R}^d\). This situation is covered, in the i.i.d. case, by Cramér’s theorem, and in more general situations, by the Gärtner-Ellis theorem. Chapter 3 contains applications of the theory developed in Chapter 2, among them the LDP for finite-state Markov chains, the hypothesis testing problem and rate distortion theory, i.e. the problem of source coding. In Chapter 4, LDP’s for families of measures on more general spaces are studied. This chapter contains the contraction principle, Varadhan’s integral lemma and large deviations for projective limits. Relations to the theory of weak convergence are also investigated. Chapter 5 contains again applications: sample path large deviations for random walk and Brownian motion, Freidlin-Wentzell theory, diffusion exit from a domain. In Chapter 6, LDP’s for empirical measures and empirical means are considered in infinite-dimensional spaces. LDP’s for empirical measures of stationary processes are established under mixing conditions. In Chapter 7, three of the applications in Chapter 2 are extended to Polish spaces: universal hypothesis testing, sampling without replacement and the Gibbs conditioning principle.

Each chapter contains historical notes and references. Comparing to the first edition (1993; Zbl 0793.60030), several sections have been added (among them Section 2.4 on concentration inequalities and Section 6.6 on the weak convergence approach to large deviations), and some subsections have been rewritten. Also, the second edition has a much larger bibliography.

Chapter 1 is devoted to the introduction and definition of a large deviation principle (LDP). Chapter 2 treats LDP’s on finite-dimensional spaces; the main example is the empirical mean of a sequence of random variables taking values in \(\mathbb{R}^d\). This situation is covered, in the i.i.d. case, by Cramér’s theorem, and in more general situations, by the Gärtner-Ellis theorem. Chapter 3 contains applications of the theory developed in Chapter 2, among them the LDP for finite-state Markov chains, the hypothesis testing problem and rate distortion theory, i.e. the problem of source coding. In Chapter 4, LDP’s for families of measures on more general spaces are studied. This chapter contains the contraction principle, Varadhan’s integral lemma and large deviations for projective limits. Relations to the theory of weak convergence are also investigated. Chapter 5 contains again applications: sample path large deviations for random walk and Brownian motion, Freidlin-Wentzell theory, diffusion exit from a domain. In Chapter 6, LDP’s for empirical measures and empirical means are considered in infinite-dimensional spaces. LDP’s for empirical measures of stationary processes are established under mixing conditions. In Chapter 7, three of the applications in Chapter 2 are extended to Polish spaces: universal hypothesis testing, sampling without replacement and the Gibbs conditioning principle.

Each chapter contains historical notes and references. Comparing to the first edition (1993; Zbl 0793.60030), several sections have been added (among them Section 2.4 on concentration inequalities and Section 6.6 on the weak convergence approach to large deviations), and some subsections have been rewritten. Also, the second edition has a much larger bibliography.

Reviewer: Nina Gantert (Berlin)

### MSC:

60F10 | Large deviations |

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

60B10 | Convergence of probability measures |