Large deviations and applications (Q2784994)

This is a very nice review of the theory of large deviations. In a concise form the reader is quickly introduced to foundations and main results of this theory. The review follows largely the logical structure of the authors [``Large deviations techniques and applications'' (1998; Zbl 0896.60013)]. In Section 6.2 the definition of large deviation principle (LDP) and some of its basic properties are given. Section 6.3 treats LDP's in finite-dimensional spaces. In Section 6.4 some methods in establishing LDP's for general spaces are discussed including e.g. contraction principle, Varadhan's lemma and the behaviour of LDP under projective limits. Section 6.5 presents sample path LDP's for random walk, Brownian motion and diffusion processes and the application to diffusion exit from a domain. Finally, Section 6.6 deals with Sanov's theorems for empirical measures of i.i.d. sequences, Markov chains and some mixing sequences. Two particular applications of LDP to statistics and statistical mechanics conclude the review.NEWLINENEWLINENEWLINEIn the bibliography leading monographs and papers are given. To complete the picture, in the introduction some other topics, which have been omitted in the review, are listed and references to respective monographs and papers are given.NEWLINENEWLINEFor the entire collection see [Zbl 0979.00017].