Approximation and Weak Convergence Methods for Random Processes with Applications to Stochastic Systems Theory

From Signal Processing, Optimization, and Control

Approximation and Weak Convergence Methods for Random Processes with Applications to Stochastic Systems Theory

By Harold Kushner





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.

This book is the sixth in The MIT Press Series in Signal Processing, Optimization, and Control, edited by Alan S. Willsky.


Out of Print ISBN: 9780262110907 287 pp. | 6 in x 9 in


$35.00 X ISBN: 9780262512183 287 pp. | 6 in x 9 in


  • The book providesan excellent development of the techniques of asymptotic analysis and weak convergence theory for stochastics dynamical systems. The chapters devoted to applications provide especially valuable illustrations of the uses of the methods.

    Gilmer Blankenship

    University of Maryland

  • An important addition to the literature of stochastic analysis.

    Mark Pinsky

    Northwestern University

  • This book is very interesting particularly in the way in which the author uses careful and precise mathematical analysis to treat problems arising in the practical fields of communication and control. Engineering applications such as adaptive filters and antenna arrays, adaptive quantifiers and phase locked loops, are handled, for the first time using Approximation and Weak convergence Method. Therefore, the author's contribution is twofold: to better understanding the phenomena behind these applications and to regorously solving the more generalized forms of these phenomena.

    Y. Bar-Ness

    AT&T-Bell Laboratories