STAT C206A / MATH C223A : Stein's method and applications (Fall 2007; Listed as "Stochastic Processes")

Instructor: Sourav Chatterjee

Class time: MWF 2:00 - 3:00

Location: 3111 Etcheverry

Scribe notes:

Lecture 1 (8/27/2007) (Introduction.)
Lecture 2 (8/29/2007) (Distances between probability measures, Integration by parts for the Gaussian measure.)
Lecture 3 (8/31/2007) (First step in Stein's method, basic examples.)
Lecture 4 (9/5/2007) (Proof of Stein's bounds.)
Lecture 5 (9/7/2007) (Proof of Stein's bounds contd.; Introduction to dependency graphs.)
Lecture 6 (9/10/2007) (Dependency graphs contd.; examples.)
Lecture 7 (9/12/2007) (Method of exchangeable pairs; examples.)
Lecture 8 (9/14/2007) (Hoeffding's combinatorial CLT using exchangeable pairs.)
Lecture 9 (9/17/2007) (Hoeffding CLT contd.; other examples.)
Lecture 10 (9/19/2007) (Concentration inequalities; first example: Efron-Stein bound.)
Lecture 11 (9/21/2007) (Concentration in first passage percolation.)
Lecture 12 (9/24/2007) (Stein's method for concentration inequalities.)
Lecture 13 (9/26/2007) (Stein's method for concentration inequalities: Two examples.)
Lecture 14 (9/28/2007) (Introduction to strong embeddings via Stein's method.)
Lecture 15 (10/1/2007) (Introduction to strong embeddings contd.)
Lecture 16 (10/3/2007) (Stein's method for strong embeddings: General plan of attack.)
Lecture 17 (10/5/2007) (Strong embeddings: The main lemma.)
Lecture 18 (10/8/2007) (Tusnády's lemma.)
Lecture 19 (10/10/2007) (Proof of KMT contd.)
Lecture 20 (10/12/2007) (Proof of KMT finished.)
Lecture 21 (10/15/2007) (Introduction to spin glasses: The Sherrington-Kirkpatrick model.)
Lecture 22 (10/17/2007) (Concentration of the partition function.)
Lecture 23 (10/19/2007) (Guerra's replica-symmetric upper bound.)
Lecture 24 (10/22/2007) (Concentration of the overlap at high temperature.)
Lecture 25 (10/24/2007) (Annealed CLTs in the S-K model via Stein's method.)
Lecture 26 (10/26/2007) (Annealed CLTs contd.)
Lecture 27 (10/29/2007) (Quenched CLTs.)
Lecture 28 (10/31/2007) (TAP equations via Stein's method.)
Lecture 29 (11/2/2007) (SK model at the critical temperature.)
Lecture 30 (11/5/2007) (A generalization of the Gaussian Poincaré inequality.)
Lecture 31 (11/7/2007) (Second order Poincaré inequalities.)
Lecture 32 (11/9/2007) (CLT for linear statistics of eigenvalues.)
Lecture 33 (11/14/2007) (Zero bias couplings.)
Lecture 34 (11/16/2007) (Size bias couplings. Introduction to interaction graphs.)

Sample LaTeX file: Lecture3.tex

Tentative list of scribes: Arnab Sen (9/5), Guy Bresler (9/7), Allan Sly (9/10), Partha Dey (9/12), Laura Derksen (9/14), Tanya Gordeeva (9/17), Chris Haulk (9/19), Maximilian Kasy (9/21), Richard Liang (9/24), Joel Mefford (9/26), Yiran Zhu (9/28), Arnab Sen (10/1), Anand Sarwate (10/3).

Partha Dey (10/5), Guy Bresler (10/8), Laura Derksen (10/10), Tanya Gordeeva (10/12), Chris Haulk (10/15), Maximilian Kasy (10/17), Partha Dey (10/19), Joel Mefford (10/22), Yiran Zhu (10/24), Richard Liang (10/26).

Arnab Sen (10/29), Anand Sarwate (10/31), Tanya Gordeeva (11/2), Laura Derksen (11/5), Anand Sarwate (11/7), Guy Bresler (11/9), Maximilian Kasy (11/14), Joel Mefford (11/16), Yiran Zhu (11/19), Chris Haulk (11/21), Richard Liang (11/26).

Course Topics

1. Stein operators, inversions, and basic lemmas.
2. Classical techniques and applications:
• The method of dependency graphs.
• Size bias and zero bias couplings.
• Exchangeable pairs.
• The diffusion approach to Stein's method.
3. Concentration inequalities using Stein's method, including:
• Basics of concentration inequalities: Ledoux, Talagrand, etc.
• The semigroup method and connection with exchangeable pairs.
• Applications to Gibbs measures and combinatorial problems.
4. Applications to random matrices, including:
• Derivation of Wigner's theorem using Stein's method.
• Central limit theorems for random matrix spectra.
5. Applications to spin glasses, including:
• An introduction to Talagrand's work.
• Derivation of the TAP equations for the SK model of spin glasses using Stein's method.
• Analysis of the critical temperature regimes of the Curie-Weiss and SK models using Stein's method.
6. Dynamic interaction graphs and recent applications.
7. Applications to strong embedding problems.
8. If time permits, discussion of the Chen-Stein method of Poisson approximation.
9. Discussion of open problems.

Prerequisite

Grading

Office Hours