Intended for a second course in stationary processes, Stationary Stochastic Processes: Theory and Applications presents the theory behind the field's widely scattered applications in engineering and science. In addition, it reviews sample function properties and spectral representations for stationary processes and fields, including a portion on stationary point processes. Features Presents and illustrates the fundamental correlation and spectral methods for stochastic processes and random fields Explains how the basic theory is used in special applications like detection theory and signal processing, spatial statistics, and reliability Motivates mathematical theory from a statistical model-building viewpoint Introduces a selection of special topics, including extreme value theory, filter theory, long-range dependence, and point processes Provides more than 100 exercises with hints to solutions and selected full solutions This book covers key topics such as ergodicity, crossing problems, and extremes, and opens the doors to a selection of special topics, like extreme value theory, filter theory, long-range dependence, and point processes, and includes many exercises and examples to illustrate the theory. Precise in mathematical details without being pedantic, Stationary Stochastic Processes: Theory and Applications is for the student with some experience with stochastic processes and a desire for deeper understanding without getting bogged down in abstract mathematics.
Some Probability and Process Background Sample space, sample function, and observables Random variables and stochastic processes Stationary processes and fields Gaussian processes Four historical landmarks Sample Function Properties Quadratic mean properties Sample function continuity Derivatives, tangents, and other characteristics Stochastic integration An ergodic result Exercises Spectral Representations Complex-valued stochastic processes Bochner's theorem and the spectral distribution Spectral representation of a stationary process Gaussian processes Stationary counting processes Exercises Linear Filters - General Properties Linear time invariant filters Linear filters and differential equations White noise in linear systems Long range dependence, non-integrable spectra, and unstable systems The ARMA-family Linear Filters - Special Topics The Hilbert transform and the envelope The sampling theorem Karhunen-Loeve expansion Classical Ergodic Theory and Mixing The basic ergodic theorem in L2 Stationarity and transformations The ergodic theorem, transformation view The ergodic theorem, process view Ergodic Gaussian sequences and processes Mixing and asymptotic independence Vector Processes and Random Fields Spectral representation for vector processes Some random field theory Exercises Level Crossings and Excursions Level crossings and Rice's formula Poisson character of high-level crossings Marked crossings and biased sampling The Slepian model Crossing problems for vector processes and fields A Some Probability Theory Events, probabilities, and random variables The axioms of probability Expectations Convergence Characteristic functions Hilbert space and random variables B Spectral Simulation of Random Processes The Fast Fourier Transform, FFT Random phase and amplitude Simulation scheme Difficulties and details Summary C Commonly Used Spectra D Solutions and Hints To Selected Exercises Some probability and process background Sample function properties Spectral and other representations Linear filters - general properties Linear filters - special topics Ergodic theory and mixing Vector processes and random fields Level crossings and excursions Some probability theory Bibliography Index
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