Approximation and Weak Convergence Methods for Random Processes, With Applications to Stochastic Systems Theory.

TLDR

The book is positioned for control and communications engineers, physicists, and probability theorists, offering a unique perspective across these fields. It develops comprehensive approximation and limit theorems for random processes, including conditions for convergence to Markov diffusion or jump processes, and introduces weak convergence techniques such as perturbed test functions, martingale methods, and direct averaging, while addressing state‑dependent noise, nonsmooth dynamics, and stability of non‑Markovian systems. By simplifying and extending older methods, the book presents powerful new techniques applicable to a wide range of limit and approximation problems, proving especially useful when complex physical models are approximated by simpler diffusion processes. It is the sixth volume in MIT Press’s Signal Processing, Optimization, and Control series, edited by Alan S.

Abstract

Control and communications engineers, physicists, and probability theorists, among others, will find this book unique. It contains a detailed development of approximation and limit theorems and methods for random processes and applies them to numerous problems of practical importance. In particular, it develops usable and broad conditions and techniques for showing that a sequence of processes converges to a Markov diffusion or jump process. This is useful when the natural physical model is quite complex, in which case a simpler approximation la diffusion process, for example) is usually made.The book simplifies and extends some important older methods and develops some powerful new ones applicable to a wide variety of limit and approximation problems. The theory of weak convergence of probability measures is introduced along with general and usable methods (for example, perturbed test function, martingale, and direct averaging) for proving tightness and weak convergence.Kushner's study begins with a systematic development of the method. It then treats dynamical system models that have state-dependent noise or nonsmooth dynamics. Perturbed Liapunov function methods are developed for stability studies of nonMarkovian problems and for the study of asymptotic distributions of non-Markovian systems. Three chapters are devoted to applications in control and communication theory (for example, phase-locked loops and adoptive filters). Smallnoise problems and an introduction to the theory of large deviations and applications conclude the book.Harold J. Kushner is Professor of Applied Mathematics and Engineering at Brown University and is one of the leading researchers in the area of stochastic processes concerned with analysis and synthesis in control and communications theory. This book is the sixth in The MIT Press Series in Signal Processing, Optimization, and Control, edited by Alan S. Willsky.