Brian Van Koten
Assistant Professor
Email: vankoten(at)math.umass.edu
Office: LGRT 1428
Website: See my personal website
here.
If you have questions or concerns, talk to me during my office hours or make an appointment. My office hours are listed on my math department directory page.
I may change my office hours during the course of the semester. Any such changes will be announced in advance in class.
Lectures will be held TTh from 2:30-3:45pm in LGRT 1322.
Week 1: Definitions of measures and sigma algebras. Construction of measures, Section 2.1-2.3 and 2.4 in Rosenthal.
Week 2: Measurable maps and random variables, Chapter 8 in Jacod and Protter. Continuity of probability, Section 3.3 in Rosenthal. Expected value and the integral, Chapter 9 in Jacod and Protter.
Week 3: Expected value and the integral, Chapter 9 in Jacod and Protter. Distributions of random variables, Section 1.2 in Durrett. Independence, Chapter 10 in Jacod and Protter.
Week 4: Independence, product measures, and the Fubini--Tonelli theorem, Chapter 10 in Jacod and Protter. Weak law of large numbers, Durrett Theorem 2.2.3. L^2 as an inner product space, first page of Chapter 22 in Jacod and Protter. Chebyshev's Inequality. Covariance and correlation. Convergence of random variables, Chapter 17 in Jacod and Protter.
Week 5: Convergence of random variables, Chapter 17 in Jacod and Protter. Borel--Cantelli Lemma, Section 3.4 in Rosenthal or Chapter 10 in Jacod and Protter. Proof of strong law of large numbers, Durrett Section 2.4.
Week 6: Proof of strong law of large numbers, Durrett Section 2.4. Weak convergence, Jacod and Protter Chapter 18 and Rosenthal Chapter 10.
Week 7: Weak convergence, Rosenthal Chapter 10.
Week 8: Characteristic Functions, Chapters 13 and 14 in Jacod and Protter. The one-dimensional normal distribution. Tightness and the Helly Selection Principle, Theorem 18.6 in Jacod and Protter or Lemma 11.1.8 and Theorem 11.1.10 in Rosenthal. Levy's Continuity Theorem, Chapter 19 in Jacod and Protter.
Week 9: Levy's Continuity Theorem, Chapter 19 in Jacod and Protter. Proof of Central Limit Theorem, Chapter 21 in Jacod and Protter. Differentiation under the integral, Proposition 9.2 in Rosenthal. Multivariate normal distribution, Chapter 16 in Jacod and Protter. Cramer--Wold device, Theorem 3.9.5 in Durrett. Proof of multivariate central limit theorem, Theorem 3.9.6 in Durrett.
Week 10: Conditional Probability and Expectation, Chapter 5.1 in Durrett, Chapter 13 in Rosenthal. See Probability by Leo Breiman for a discussion of E[X|Y=y] and its relationship with E[X|Y].
Week 11: Conditional Probability and Expectation, Chapter 5.1 in Durrett, Chapter 13 in Rosenthal. Martingales, Chapter 5.2 in Durrett.
Week 12 and 13: Martingales, the Martingale Convergence Theorem, Doob's inequality and convergence in Lp, Uniform integrability and convergence in L1, Chapter 5 in Durrett.
Homework will be assigned after every class. All assigments will be posted below. Problems assigned during each week will be due the following week on Thursday.
No late homework will be accepted. Instead, when the total course grade is computed, the lowest homework score will be dropped.
I encourage you to discuss the homework with your classmates and to come to my office hours with your questions.
Assignments due 9/13: HW1-1 and HW1-2.
Assignments due 9/20: HW2-1 and HW2-2.
Assignments due 9/27: HW3-1 and HW3-2.
Assignments due 10/4: HW4-1 and HW4-2.
Assignments due 10/11: HW5-1 and HW5-2.
Assignments due 10/18: HW6-1.
Assignments due 10/29: HW7-1 and HW7-2.
Assignments due 11/8: HW8-1 and HW8-2.
Assignments due 11/15: HW9-1 and HW9-2.
Assignments due 11/29: HW10-1.
Assignments due 12/6: HW11-1.
Assignments due 12/12: HW12-1.
The exam is due Friday, October 26 by 3:00pm in my mailbox or email.
Please do not discuss the problems on the midterm with anyone. You may consult any of the recommended texts, but please do not search the web for solutions.
The final exam will be designed to resemble the Ph.D. program's qualifying exam in probability.
If for any reason you will not be able to take either exam at the specified time, let me know as soon as possible.
If you suspect that a mistake has been made in the grading of your work, please point it out to me no more than two weeks after the work was returned. I will not consider complaints if more than two weeks have passed. Please do not ask the grader to change homework grades; come to me with any disputes.
Jeffrey S. Rosenthal, A First Look at Rigorous Probability Theory, 2nd edition. World Scientific 2006. ISBN-10: 9812703713 ISBN-13: 978-9812703712
Other Useful TextsRick Durrett, Probability: Theory and Examples. Available for free here.
This excellent book contains many interesting and important theorems, examples, and problems not covered in the official text. If you find the official text boring and wonder how the basic ideas are applied in practice, read this book.
Sidney Resnick, A Probability Path.
If you find the official text too concise, you might prefer the exhaustively detailed explanations in this book.
Jean Jacod and Philipp Protter, Probability Essentials.
This outstanding book is similar to the assigned text, but the explanations may be slightly less concise and slightly less technical.
Axioms of probability and the construction of probability spaces.
Random variables, integration, convergence of sequences of random variables, and the law of large numbers.
Gaussian random variables, characteristic and moment generating functions, and the central limit theorem.
Conditional expectation, the Radon--Nikodym theorem, and martingales.
Introductory courses in probability and analysis. No knowledge of measure theory will be assumed.
The total course score will be computed according to the following formula: 34% Homework + 33% Midterm + 33% Final.
If your score is in the top third, you will receive an A. If your score is in the middle third, you will receive at least a B, but possibly a higher grade. If I am convinced that all students understand the basic principles of the course, I will not give any C's, D's, or F's. I will make the distribution of scores on the midterm and the final exam known.