Introduction To Random Processes

Rigorous exposition suitable for elementary instruction. Covers measure theory, axiomatization of probability theory, processes with independent increments, Markov processes and limit theorems for random processes, more. A wealth of results, ideas, and techniques distinguish this text. Introduction. Bibliography. 1969 edition.

The book covers basic concepts such as random experiments, probability axioms, conditional probability, and counting methods, single and multiple random variables (discrete, continuous, and mixed), as well as moment-generating functions, characteristic functions, random vectors, and inequalities; limit theorems and convergence; introduction to Bayesian and classical statistics; random processes including processing of random signals, Poisson processes, discrete-time and continuous-time Markov chains, and Brownian motion; simulation using MATLAB and R.

This text on stochastic processes and their applications is based on a set of lectures given during the past several years at the University of California, Santa Barbara (UCSB). It is an introductory graduate course designed for classroom purposes. Its objective is to provide graduate students of statistics with an overview of some basic methods and techniques in the theory of stochastic processes. The only prerequisites are some rudiments of measure and integration theory and an intermediate course in probability theory. There are more than 50 examples and applications and 243 problems and complements which appear at the end of each chapter. The book consists of 10 chapters. Basic concepts and definitions are pro vided in Chapter 1. This chapter also contains a number of motivating ex amples and applications illustrating the practical use of the concepts. The last five sections are devoted to topics such as separability, continuity, and measurability of random processes, which are discussed in some detail. The concept of a simple point process on R+ is introduced in Chapter 2. Using the coupling inequality and Le Cam's lemma, it is shown that if its counting function is stochastically continuous and has independent increments, the point process is Poisson. When the counting function is Markovian, the sequence of arrival times is also a Markov process. Some related topics such as independent thinning and marked point processes are also discussed. In the final section, an application of these results to flood modeling is presented.

Breaking with the traditional treatment of random processes in engineering On the surface, Introduction to Random Processes in Engineering is simply a first-rate textbook for senior or first-year graduate engineering courses in stochastic processes. A closer look, however, reveals an innovative book—rich with examples and commonsense explanations—that demystifies theories, eliminates ambiguities, and provides a solid, up-to-date introduction to this important subject. Departing from the classical texts of the sixties and seventies in its coverage of random signals and data processing, Introduction to Random Processes in Engineering addresses the latest advances in communication, control engineering, and signal processing by allowing all processes to be multidimensional with an emphasis on discrete-time processes and systems. Unlike current texts, this volume provides a strong mathematical perspective for its engineering topics without getting bogged down in technicalities. It employs mathematics to achieve clarity and precision, and at times even uses the theorem/proof style to emphasize mathematical fine points. This approach is particularly advantageous when dealing with random data, and when building an understanding of the many computer programs routinely used, its theoretical principles, and the results it generates. Assuming a senior-level background in probability theory and some acquaintance with linear systems and signals, the book provides: A review chapter of the formulas used later in the book Illustrative examples Emphasis in simulation techniques Problems accompanying each chapter that often introduce the student to other relevant material Notes and comments following each chapter that encourage additional reading as well as historical explorations in the field Tips for using the material at various levels of instruction With its logical and systematically ordered presentation of the material, as well as its fresh approach, Introduction to Random Processes in Engineering is both a superior textbook and a valuable reference for practicing engineers and researchers in the field.

This book concentrates on some general facts and ideas of the theory of stochastic processes. The topics include the Wiener process, stationary processes, infinitely divisible processes, and Ito stochastic equations. Basics of discrete time martingales are also presented and then used in one way or another throughout the book. Another common feature of the main body of the book is using stochastic integration with respect to random orthogonal measures. In particular, it is used forspectral representation of trajectories of stationary processes and for proving that Gaussian stationary processes with rational spectral densities are components of solutions to stochastic equations. In the case of infinitely divisible processes, stochastic integration allows for obtaining arepresentation of trajectories through jump measures. The Ito stochastic integral is also introduced as a particular case of stochastic integrals with respect to random orthogonal measures. Although it is not possible to cover even a noticeable portion of the topics listed above in a short book, it is hoped that after having followed the material presented here, the reader will have acquired a good understanding of what kind of results are available and what kind of techniques are used toobtain them. With more than 100 problems included, the book can serve as a text for an introductory course on stochastic processes or for independent study. Other works by this author published by the AMS include, Lectures on Elliptic and Parabolic Equations in Holder Spaces and Introduction to the Theoryof Diffusion Processes.