*(English)*Zbl 0793.60030

Various aspects of the theory and applications of large deviations principle (LDP) are considered. The book consists of seven chapters and Appendix.

Chapter 1 is devoted to the introduction and formulation of LDP. The finite-dimensional case is studied in Chapter 2. The main example is the empirical mean of a sequence of random variables taking values in ${\mathbb{R}}^{d}$. Sanov’s and Cramer’s theorems are proved. The applications of the theory developed in Chapter 2 are presented in Chapter 3. The LDPs associated with the finite state irreducible Markov chains are derived and the asymptotic size of long rare segments in random walks is found. The asymptotics of the probability of error in hypothesis testing problems are analysed.

The LDPs for families of measures on general spaces are studied in Chapter 4. Relations between the topological structure of the space, the existence and uniqueness of LDP are explored. The transfer of LDP from one space to another is investigated. In Chapter 5 the probability that a path of a random process hits a particular set is found. The cases of random walk, Brownian motion and diffusion, the Frejdlin-Venttsel’ theory, are considered. The main results of Chapter 2 are generalized in Chapter 6 moving away from finite-dimensional case. The LDP for stationary processes satisfying a certain mixing condition is established as well. In Chapter 7 the applications considered in Chapters 2 and 3 are extended to the case of Polish space. Sanov’s theorem and projective limit approach is treated.

The appendix presents basic accompanying notions and facts making the book more self-contained. Each chapter contains historical notes of references.