Math 218a / Stat 205A. Probability Theory.

Fall 2017, TTh 9:30a-11:00a in Etcheverry 3108.

GENERAL INFORMATION. (## = berkeley dot edu)
Please include "MATH218A/STAT205A" and your SID in the subject line of all emails.
Instructor: Nike Sun (nikesun at ##). Office hours Tue 2:15p-4:15p in Evans 389.
GSI: Zsolt Bartha (bartha at ##). Office hours Mon 11:30a-1:00p and Wed 1:30p-3:00p in Evans 428.
Discussion section Fri 11:00a-12:00n in Evans 330.

Some materials, including homework assignments, will be posted to bcourses. If you cannot access the site, email the instructor with your @berkeley.edu address.

REFERENCES. References marked * are available electronically through lib.berkeley.edu.
Main reference:
*Probability: Theory and Examples (4th ed.) by R. Durrett (PTE).
Additional references:
Lecture notes by A. Dembo, for a similar course taught at Stanford.
Probability and Measure (3rd ed.) by P. Billingsley.
*Convergence of Probability Measures by P. Billingsley.
*Real Analysis and Probability by R. M. Dudley.
*Measure Theory by P. R. Halmos.
Foundations of Modern Probability by O. Kallenberg.
Analysis by E. H. Lieb and M. Loss.
Probability on Trees and Networks by R. Lyons and Y. Peres (link).
Complex Analysis by E. Stein and R. Shakarchi.

HOMEWORK. Please be considerate of the grader: write solutions neatly, and staple.
At the top of the first page, write clearly the following information: your name, student ID number, Berkeley email address, the course name (MATH218A/STAT205A), and the names of any other students with whom you discussed the assignment. Please also indicate if you are currently waitlisted.
Collaboration policy. You should first attempt to solve homework problems on your own. If you are having trouble, you may discuss with others. You are expected to write down your solutions alone.

GRADING. Your final grade will consist of 50% homework + 50% final.
No late homework will be accepted. Lowest two homework scores will be dropped.
Missing the final will automatically result in an F grade; there will be no rescheduling (early/late/repeat). All students will be graded under the same scheme, regardless of late enrollment.

SCHEDULE. Any reading listed from PTE is mandatory; from other references is optional.

• Lecture 1 (08/24) measure spaces; non-measurable sets. PTE 1.1.
• Lecture 2 (08/29) measures on the real line: from semialgebra to algebra. PTE A.1.
  Homework 1 posted, due 09/05.
• Lecture 3 (08/31) measures on the real line: from algebra to sigma-field. PTE A.2.
• Lecture 4 (09/05) random variables; integration. PTE 1.2-1.4.
  Homework 2 posted, due 09/12.
• Lecture 5 (09/07) integral convergence theorems; change of variables formula. PTE 1.5-1.6.
• Lecture 6 (09/12) product of two measures; Tonelli and Fubini theorems; product of infinitely many probability measures. PTE 1.7, Kallenberg 5.17-5.18.
• Lecture 7 (09/14) regularity of measures; Polish spaces; Kolmogorov extension theorem. PTE A.3, Dudley 7.1, Kallenberg 5.16.
  Homework 3 posted, due 09/26.
• Lecture 8 (09/19) weak law of large numbers (Chebychev's inequality, truncation); St. Petersburg paradox; strong law of large numbers (truncation, subsequence method). PTE 2.1-2.2 and 2.4.
• Lecture 9 (09/21) large deviations for sums of i.i.d. random variables; exponentially tilted measures; Legendre dual. PTE 2.6.
• Lecture 10 (09/26) characteristic functions; Fourier transform (on discrete cube, on real line); unitarity (Plancherel, Parseval). PTE 3.3, Lieb & Loss Ch. 5, Stein & Shakarchi Ch. 4.
• Lecture 11 (09/28) (lecture given by Zsolt) convergence of random series. PTE 2.3 and 2.5.
  Homework 4 posted, due 10/05.
• Lecture 12 (10/03) Fourier inversion, weak convergence, CLT for i.i.d. sequences. PTE 3.2, 3.3.1-3.3.3, 3.4.1. Lieb & Loss Ch. 5.
• Lecture 13 (10/05) Lindeberg--Feller CLT (example: random permutation cycle counts), Poisson limit theorem, weak convergence on metric spaces. PTE 3.2, 3.4.2, 3.6.1. Convergence of Probability Measures (Billingsley) Ch. 1.
  Homework 5 posted, due 10/12.
• Lecture 14 (10/10) measure-theoretic independence; tail sigma-field & Kolmogorov zero-one law; exchangeable sigma-field & Hewitt--Savage zero-one law. PTE 2.1, 4.1.
• Lecture 15 (10/12) stopping times; recurrence & transience for integer lattice SRW. PTE 4.1-4.2.
  Homework 6 posted, due 10/19.
• Lecture 16 (10/17) euclidean random walks; Chung--Fuchs recurrence criterion. PTE 4.2.
• Lecture 17 (10/19) Chung--Fuchs recurrence criterion (examples); integer SRW: reflection and ballot theorem. PTE 4.2, 4.3.
• Lecture 18 (10/24) arcsine laws (integer SRW, Sparre-Anderson). PTE 4.3, Kallenberg Ch. 8.
• Lecture 19 (10/26) renewal processes; size-biasing; ladder heights and occupation measures for transient random walks on the line. PTE 4.4, Kallenberg Ch. 8.
  Homework 7 posted, due 11/03.
• Lecture 20 (10/31) conditional expectation. PTE 5.1.
• Lecture 21 (11/02) definition of (sub)martingales and connection to (sub)harmonic functions -- graph Laplacian, Dirichlet boundary value problem, (very brief) introduction to electrical networks. PTE 5.2, Lyons & Peres 2.1.
• Lecture 22 (11/09) (lecture given by Zsolt) martingales: upcrossing inequality, convergence, Doob decomposition. PTE 5.2.
  Homework 8 posted, due 11/17.
• Lecture 23 (11/14) martingales: Doob's inequality, L^p maximal inequality, L^p convergence (example: Galton--Watson branching processes). PTE 5.3-5.4.
• Lecture 24 (11/16) martingales: uniform integrability and L^1 convergence; Doob martingales; Lévy convergence theorem and zero-one law. PTE 5.5.
  Homework 9 posted, due 11/28.
• Lecture 25 (11/21) martingales: optional stopping theorems (examples: asymmetric SRW, patterns in random strings). PTE 5.7.
• Lecture 26 (11/28) reverse martingales (examples: Hewitt--Savage, SLLN, de Finetti). PTE 5.6.
• Lecture 27 (11/30) martingale concentration inequalities (last lecture). C. McDiarmid, "On the method of bounded differences" Surveys in Combinatorics 141.1 (1989): 148-188.
  Homework 10 (last homework) posted, due 12/07.
• The final exam is on Wednesday December 13, 11:30-2:30, Le Conte Hall Room 3 (registrar).