MATHEMATICS 565A
Discrete Time Stochastic Processes

SPRING 2007

Overview. Using our background in probability theory, for this semester, we will develop ideas in discrete time stochastic processes. We begin with the basic concepts for stochastics and then go on to examine a variety of processes - random walks, renewal sequences, martingales, Markov processes and stationary processes.

We will not be using a text, but rather notes will be posted on-line.

Day to Day Operations. The course meets for lecture each Tuesday and Thursday from 9:30 to 10:45 in the Mathematics Building Room 501.

If you need to contact me, write me electronically at jwatkins@math.arizona.edu, call me at 621-5245, or drop by my office Math 520. My office hours are in Math 520 from 2 to 3, Monday and Tuesday and from 12 to 1 on Thursdays in Math East 145.

Evaluation of Students. We will evaluate your work in the course through your performance on homework assignments. The notes have a large number of exercises - - they will be graded at a point per problem written or presented to a maximum of 60 points. The first 20 points must be completed by February 20th, the first 40 by April 1st. I will also give a take home final worth 20 points. A student may choose to replace the final with a report.

We will be using a variety of sources,

• Probability: Theory and Examples by Richard Durrett
• Probability with Martingales by David Williams
• Markov Processes: Characterization and Convergence by Stewart N. Ethier and Thomas G. Kurtz
• Probability by Leo Brieman
• Diffusions, Markov Processes, and Martingales, Volume I by David Williams
• A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray
• An Introduction to Probability Theory and Its Applications, Volumes I and II by William Feller

Here is an approximate course syllabus

• Basic concepts - conditional expectation, filtrtions and stopping times
• Random walks - recurrence and transience, the role of dimension
• Renewal sequences and the renewal theorem
• Martingales
  • Upcrossing inequality
  • Optional sampling theorem
  • Inequalities and convergence
• Markov chains
  • Strong Markov property
  • Classification of states
  • Recurrent chains and statinary measures
  • Transient chains and asymptotic behavior
• Stationary processes
  • Birkhoff's ergodic theorem
  • Ergodicity and mixing
  • Entropy

Joe Watkins

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