Statistics 205B: Probability Theory. Spring 2003. Jim Pitman

Incomplete list of typos in the text: pdf file

• Homework 1 due 1/28/03. From Chapter 5, 1.6 (p 284), 1.7 (p 285) 2.5 , 2.6 , 2.10 (p 290) 2.11, 2.12 (p 291)

• Homework 2 due 2/4/03. 7 of the 10 exercises in Sec 5.3

• Homework 3 due 2/11/03. 8 of the 12 exercises in Sec 5.4

• Homework 4 due 2/18/03. 5.1 (p 317), 5.2 (p 319), 5.4 , 5.5 (p 323), 5.6 (p. 324), 5.7

• Homework 5 due 2/25/03. 7 exercises from Sec 5.6

• Homework 6 due 3/4/03. 7 exercises from chapter 6.

• Homework 7 due 3/11/03. exercises from 6.6 and 6.7

• Homework 8 due 3/18/03. exercises from 7.1 and 7.2

• Homework 9 due 4/1/03. exercises from 7.3 and 7.4

• Homework 10 due 4/8/03. exercises from 7.5

• Homework 11 due 4/15/03. 7 exercises from 7.6, 7.7, 7.8 and the following: find the probability that Brownian bridge crosses the straight line from (0,a) to (1,b).

2  Lecture Notes

• Lecture 1. Markov chain transition matrices. .ps file .pdf file

• Lecture 2. Hitting probabilities. .ps file .pdf file

• Lecture 3. Recurrence and Transience. .ps file .pdf file

• Lecture 4. Stationary distributions. .ps file .pdf file

• Lecture 5. Asymptotic Behaviour. .ps file .pdf file

• Lecture 6. Proof of the limit theorem. .ps file .pdf file

• Lecture 7. Examples. .ps file .pdf file

• Lecture 8. Continuous time chains. .ps file .pdf file

• Lecture 9. Stationary processes. .ps file .pdf file

• Lecture 10. The ergodic theorem. .ps file .pdf file

• Lecture 11. The range of a random walk. .ps file .pdf file

• Lecture 12. The subadditive ergodic theorem. .pdf file

• Lectures 13-14. Construction of Brownian Motion .ps file .pdf file

• Lecture 15. Basic properties of Brownian Motion. .ps file .pdf file

• Lecture 16. The Strong Markov Property. .ps file .pdf file

• Lecture 17. Hitting times and the Reflection Principle. .ps file .pdf file

• Lecture 18. The zero set of Brownian motion. .ps file .pdf file

• Lecture 19. Brownian martingales. .ps file .pdf file

• Lecture 20. Embedding of random walks in Brownian motion. .ps file .pdf file

• Lecture 21. Donsker's Theorem. See text.

• Lecture 22. Brownian Bridge .ps file .pdf file

• Lecture 23. Stochastic calculus for continuous martingales I. .ps file .pdf file

• Lecture 24. Stochastic calculus for continuous martingales II. .ps file .pdf file

• Lecture 25. Connections with partial differential equations. .ps file .pdf file

• Lecture 26. Lévy Processes. .ps file .pdf file

• Lecture 27. Feller Processes .ps file .pdf file

• Lecture 28. Finite State Markov Chains .pdf file