Stochastic Processes (Wiley Classics Library)
The theory of stochastic processes has developed so much in the last twenty years that the need for a systematic account of the subject has been felt, particularly by students and instructors of probability. This book fills that need. While even elementary definitions and theorems are stated in detail, this is not recommended as a first text in probability and there has been no compromise with the mathematics of probability. Since readers complained that omission of certain mathematical detail increased the obscurity of the subject, the text contains various mathematical points that might otherwise seem extraneous. A supplement includes a treatment of the various aspects of measure theory. A chapter on the specialized problem of prediction theory has also been included and references to the literature and historical remarks have been collected in the Appendix.
Why Read This Book
You should read Doob's Stochastic Processes if you need a rigorous, measure-theoretic foundation for modeling and reasoning about random signals encountered in DSP, communications, and radar. You will learn the precise theorems and proof techniques behind martingales, stationary processes, Gaussian processes, and prediction theory so you can apply them confidently to spectral analysis, estimation, and advanced stochastic modeling.
Who Will Benefit
Advanced graduate students, researchers, and engineers with strong mathematical background who need a deep theoretical grounding in stochastic processes to support work in statistical signal processing, communications, or radar.
Level: Advanced — Prerequisites: Undergraduate real analysis and probability (measure-theoretic probability recommended); familiarity with Lebesgue integration, linear algebra, and multivariable calculus. (Doob provides a measure-theory supplement, but prior exposure greatly helps.)
Key Takeaways
- Understand the measure-theoretic construction of stochastic processes and finite-dimensional distributions
- Apply martingale theory and convergence theorems to analyze stochastic algorithms and adaptive filters
- Derive and use spectral representation theorems for stationary processes relevant to FFT-based spectral analysis
- Use prediction theory (Wiener–Kolmogorov style) to formulate optimal linear predictors for time-series and communication signals
- Characterize Gaussian processes and Brownian motion for noise modeling in radar, audio, and communications
- Analyze sample-path properties, separability, and regularity needed when modelling continuous-time random signals
Topics Covered
- Preface and introduction
- Measure theory supplement: sigma-algebras, measures, integration
- Probability measures and random variables
- Finite-dimensional distributions and construction of processes
- Processes with independent increments and Lévy processes
- Markov processes and transition probabilities
- Martingales: definitions, convergence, and optional sampling
- Stationary processes and spectral representation
- Gaussian processes and Brownian motion
- Prediction theory for stationary processes
- Sample function properties, separability, and regularity
- Appendices, bibliographic notes, and references
How It Compares
Compared with engineering-oriented texts like Papoulis' Probability and Random Processes, Doob is far more rigorous and theorem-driven; compared with Karatzas & Shreve's SDE treatment, Doob emphasizes classical martingale and prediction theory rather than stochastic differential equations.
