Stationary Stochastic Processes: Theory and Applications

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-Loève 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