# 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

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

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