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This text defines a variety of non-Gaussian processes, develops methods for generating realizations of non-Gaussian models, and provides methods for finding probabilistic characteristics of the output of linear filters with non-Gaussian inputs.
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.
The conclusions of the report are: (1) The expansion coefficients of some common representations of random processes will be independent only when the process is Gaussian; (2) Processes representable on a particular interval in terms of a denumerable sequence of independent random variables will often have sample function properties similar to those of the Gaussian process; (3) The quadratic variation of non-Gaussian processes with sufficiently smooth cumulants is constant for a given interval; (4) The quadratic variation of a non-Gaussian linear process equals the sum of the squares of its jump discontinuities; (5) There is a class of sequences of functionals, say T sub N(x(t)), such that l.i.m. T sub N equals the quadratic variation of the processes in (3). (6) The necessary and sufficient condition for singular detection of a sure signal in Gaussian noise is sufficient for singularity when the noise is any mean square continuous process. (7) Regularity or singularity for signals depending on a random parameter, gamma, is implied by regularity or singularity for signals corresponding to each possible value of gamma when gamma has a discrete distribution or the noise is Gaussian. (8) Singular estimation of certain parameters is sometimes possible under the conditions of singular detections. (9) Some of the spectral conditions which imply singularity for Gaussian random processes continue to imply singularity for non-Gaussian processes with sufficiently smooth cumulants. (10) For other non-Gaussian processes, spectral conditions are irrelevant.