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.

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

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