Based on the Cramer-Chernoff theorem, which deals with the "rough" logarithmic asymptotics of the distribution of sums of independent, identically random variables, this work primarily approaches the extensions of this theory to dependent and, in particular, non-Markovian cases on function spaces. Recurrent algorithms of identification and adaptive control form the main examples behind the large deviation problems in this volume. The first part of the book exploits some ideas and concepts of the martingale approach, especially the concept of the stochastic exponential. The second part covers Freidlin's approach, based on the Frobenius-type theorems for positive operators, whuch prove to be effective for the cases in consideration. The book should be of value and interest to scientists in the field of probability, statistics and electrical engineering, as well as physicists dealing with statistical mechanics.
Part 1 Introduction to large deviations: Cramer-type results (the classical Cramer theorem; the extensions of Cramer's theorem); large deviations on the space of probability measures; application to statistical mechanics; basic large deviations concepts; large deviations for sums of independent and identically distributed variables in function space; applications to recursive estimation and control theory. Part 2 Large deviations for non-Markovian recursive scheme with additive "white noise". Part 3 Large deviation for the recursive scheme with stationary disturbances: large deviations for the sums of stationary; large deviations for recursive scheme with the Wold-type disturbances. Part 4 Generalization of Cramar's theorem: large deviations for sums of stationary sequences; large deviations for sums of semimartingales. Part 5 Mixing for Markov processes: definitions; main results; preliminary results; proofs of theorems 5.1-5.6; mixing coeficients for recursive procedure. Part 6 The averaging principle for some recursive schemes. Part 7 Normal deviations. Part 8 Large deviations for Markov processes: examples; Markovian noncompact case; auxiliary results; proofs of theorems 8.6-8.8; proof of theorem 8.9. Part 9 Large deviations for stationary processes: compact nonsingular case; noncompact nonsingular case. Part 10 Large deviations for empirical measures: Markov chain with Doeblin-type condition; noncompact Markov case; stationary compact case; stationary noncompact case. Part 11 Large deviations for empirical measures: compact case; noncompact case.