Three-Toed Sloth

Lecture Notes on Stochastic Processes (Advanced Probability II)

I've started putting the notes for my lectures on stochastic processes (36-754) online at the course homepage.

Contents

Table of contents, which gives a running list of definitions, lemmas, theorems, etc. This will be updated with each new lecture.

Lecture 1 (16 January)

Definition of stochastic processes, examples, random functions

Lecture 2 (18 January)

Finite-dimensional distributions (FDDs) of a process, consistency of a family of FDDs, theorems of Daniell and Kolmogorov on extending consistent families to processes

Lecture 3 (20 January)

Probability kernels and regular conditional probabilities, extendings finite-dimensional distributions defined recursively through kernels to processes (the Ionescu Tulcea theorem).

Homework Assignment 1 (due 27 January)

Exercise 1.1; Exercise 3.1. Solutions.

Lecture 4 (23 January)

One-paramater processes and their representation by shift-operator semi-groups.

Lecture 5 (25 January)

Three kinds of stationarity, the relationship between strong stationarity and measure-preserving transformations (especially shifts).

Lecture 6 (27 January)

Reminders about filtrations and optional times, definitions of various sorts of waiting times, and Kac's Recurrence Theorem.

Homework Assigment 2 (due 6 February)

Exercise 5.3; Exercise 6.1; Exercise 6.2. Solutions

Lecture 7 (30 January)

Kinds of continuity, versions of stochastic processes, difficulties of continuity, the notion of a separable random function.

Lecture 8 (1 February)

Existence of separable modifications of stochastic processes, conditions for the existence of measurable, cadlag and continuous modifications.

Lecture 9 (3 February)

Markov processes and their transition-probability semi-groups.

Lecture 10 (6 February)

Markov processes as transformed IID noise; Markov processes as operator semi-groups on function spaces.

Lecture 11 (8 February)

Examples of Markov processes (Wiener process and the logistic map). Overlaps with solutions to the second homework assignment.

10 February

Material from section 2 of lecture 10, plus an excursion into sofic processes.

Lecture 12 (13 February)

Generators of homogeneous Markov processes, analogy with exponential functions.

Lecture 13 (15 February)

The strong Markov property and the martingale problem.

Homework Assignment 3 (due 20 February)

Exercises 10.1 and 10.2

Lecture 14 (17, 20 February)

Feller processes, and an example of a Markov process which isn't strongly Markovian.

Lecture 15 (24 February, 1 March)

Convergence in distribution of cadlag processes, convergence of Feller processes, approximation of differential equations by Markov processes.

Lecture 16 (3 March)

Convergence of random walks to Wiener processes.

Homework Assignment 4 (due 13 March)

Exercise 16.1, 16.2 and 16.4.

Lecture 17 (6 March)

Diffusions, Wiener measure, non-differentiability of almost all continuous curves.

Lecture 18 (8 March)

Stochastic integrals: heuristic approach via Euler's method, rigorous approach.

Lecture 19 (20, 21, 22 and 24 March)

Examples of stochastic integrals. Ito's formula for change of variables. Stochastic differential equations, existence and uniqueness of solutions. Physical Brownian motion: the Langevin equation, Ornstein-Uhlenbeck processes.

Lecture 20 (27 March)

More on SDEs: diffusions, forward (Fokker-Planck) and backward equations. White noise.

Lecture 21 (29, 31 March)

Spectral analysis; how the white noise lost its color. Mean-square ergodicity.

Lecture 22 (3 April)

Small-noise limits for SDEs: convergence in probability to ODEs, and our first large-deviations calculations.

Lecture 23 (5 April)

Introduction to ergodic properties and invariance.

Lecture 24 (7 April)

The almost-sure (Birkhoff) ergodic theorem.

Lecture 25 (10 April)

Metric transitivity. Examples of ergodic processes. Preliminaries on ergodic decompositions.

Lecture 26 (12 April)

Ergodic decompositions. Ergodic components as minimal sufficient statistics.

Lecture 27 (14 April)

Mixing. Weak convergence of distribution and decay of correlations. Central limit theorem for strongly mixing sequences.

Lecture 28 (17 April)

Introduction to information theory. Relations between Shannon entropy, relative entropy/Kullback-Leibler divergence, expected likelihood and Fisher information.

Lecture 29 (24 April)

Entropy rate. The asymptotic equipartition property, a.k.a. the Shannon-MacMillan-Breiman theorem, a.k.a. the entropy ergodic theorem. Asymptotic likelihoods.

Lecture 30 (26 April)

General theory of large deviations. Large deviations principles and rate functions; Varadhan's Lemma. Breeding LDPs: contraction principle, "exponential tilting", Bryc's Theorem, projective limits.

Lecture 31 (28 April)

IID large deviations: cumulant generating functions, Legendre's transform, the return of relative entropy. Cramer's theorem on large deviations of empirical means. Sanov's theorem on large deviations of empirical measures. Process-level large deviations.

Lecture 32 (1 May)

Large deviations for Markov sequences through exponential-family densities.

Lecture 33 (2 May)

Large deviations in hypothesis testing and parameter estimation.

Lecture 34 (3 May)

Large deviations for weakly-dependent sequences (Gartner-Ellis theorem).

Lecture 35 (5 May)

Large deviations of stochastic differential equations in the small-noise limit (Freidlin-Wentzell theory).

References

The bibliography, currently confined to works explicitly cited.

Everything to date

In the staggeringly-unlikely event that anyone wants to keep track of the course by RSS, this should do the trick.

Enigmas of Chance; Corrupting the Young