Large Deviations for Discrete-Time Processes with Averaging

01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information. This book is mainly based on the Cramér--Chernoff renowned theorem, which deals with the 'rough' logarithmic asymptotics of the distribution of sums of independent, identically distributed random variables. The authors approach primarily the extensions of this theory to dependent, and in particular, nonmarkovian 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 of the book covers Freindlin's approach, based on the Frobenius-type theorems for positive operators, which prove to be effective for the cases in consideration.

Table of Contents:
INTRODUCTION TO LARGE DEVIATIONS Cramér-type results (the classical Cramér theorem; the extensions of Cramér'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 LARGE DEVIATIONS FOR NON-MARKOVIAN RECURSIVE SCHEME WITH ADDITITIVE 'WHITE NOISE' 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 GENERALIZATION OF CRAMÉR'S THEOREM Large deviations for sums of stationary sequence Large deviations for sums of semimartingales MIXING FOR MARKOV PROCESSES Definitions Main results Preliminary results Proofs of theorems 5.1--5.6 Mixing coefficients for recursive procedure THE AVERAGING PRINCIPLE FOR SOME RECURSIVE SCHEMES NORMAL DEVIATIONS LARGE DEVIATIONS FOR MARKOV PROCESSES Introduction Examples Markovian noncompact case Auxiliary results Proofs of theorems 8.6--8.8 Proof of theorem 8.9 LARGE DEVIATIONS FOR STATIONARY PROCESSES Compact nonsingular case Noncompact nonsingular case LARGE DEVIATIONS FOR EMPIRICAL MEASURES Introduction Markov chain with Doeblin-type condition Noncompact Markov case Stationary compact case Stationary noncompact case LARGE DEVIATIONS FOR EMPIRICAL MEASURES Compact case Noncompact case