Almost None of the Theory of Stochastic Processes

A Course on Random Processes, for Students of Measure-Theoretic Probability, with a View to Applications in Dynamics and Statistics

by Cosma Rohilla Shalizi

with Aryeh Kontorovich

Snapshot of a non-stationary spatiotemporal stochastic process (the Greenberg-Hastings model)

This is a book-in-progress; I hope you'll find it useful, but I'm certain that it can be improved, and that it contains errors. Bug reports are very much appreciated!

This book began as the lecture notes for 36-754, a graduate-level course in stochastic processes. The official textbook for the course was Olav Kallenberg's excellent Foundations of Modern Probability, which explains the references to it for background results on measure theory, functional analysis, the occasional complete punting of a proof, etc.

At some point, I'll explain why I felt compelled to produce Yet Another Textbook on Stochastic Process.

Brief Contents

I: Stochastic Processes in General
Basics; Building Processes; Building Processes by Conditioning.
II: One-Parameter Processes in General
One-Parameter Processes; Stationary Processes; Random Times; Continuity.
III: Markov Processes
Markov Processes; Markov Characterizations; Markov Examples; Generators; the Strong Markov Property and Martingale Problems; Feller Processes; Convergence of Feller Processes; Convergence of Random Walks.
IV: Diffusions and Stochastic Calculus
Diffusions and the Wiener Process; Stochastic Integrals and Stochastic Differential Equations; Spectral Analysis and White Noise; Small-Noise SDEs.
V: Ergodic Theory
Mean-Square Ergodicity; Ergodic Properties and Ergodic Limits; the Almost-Sure Ergodic Theorem; Ergodicity and Metric Transitivity; Ergodic Decomposition; Mixing; Asymptotic Distributions.
VI: Information Theory
Entropy and Divergence; Rates and Equipartition; Information Theory and Statistics.
VII: Large Deviations
Large Deviations Basics; IID Large Deviations; Large Deviations for Markov Sequences; the Gartner-Ellis Theorem; Large Deviations for Stationary Sequences; Large Deviations in Inference; Freidlin-Wentzell Theory.
VIII: Measure Concentration, by Aryeh (Leonid) Kontorovich
IX: Partially Observable Processes
Hidden Markov Models; Stochastic Automata; Predictive Representations.
X: Applications
Reminders of definitions and results (without proof) from analysis, measure theory, Laplace transforms, etc.

Available Files

The latest version of the text is 0.1.1 (3 December 2007). You can download the full text (PDF, 3.8 M, 331 pp.), or the PDF table of contents. (Chapters marked "[w]" in the table of contents still have to be written.)
Page made 2 December 2007; last changed 3 December 2007