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

Appendices

Reminders of definitions and results (without proof) from analysis, measure theory, Laplace transforms, etc.

Available Files