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Math 672/673

Theory of Probability

Winter/Spring 2023. MWF 10-11am.

Catalog description: Measure and integration, probability spaces, laws of large numbers, central-limit theory, conditioning, martingales, random walks.


Homework (30%)

There will be weekly homework, due Friday of each week on Canvas. Please submit as PDF, ideally using LaTeX or some similar typesetting program. Each problem should be written in “Theorem, Proof” form, where you state what you are proving as a theorem (you may rephrase the assigned questions to express it as a theorem if necessary) and then provide a proof in its own clearly marked section.

You will also be asked to review assignments of other students as a “referee”. This will require you to read and provide constructive input on the proofs of your peers. In general, for each problem you should either “accept” the proved theorem as correct with a well-written proof, “revise and resubmit” if there are minor errors that can be fixed easily or the writing lacks clarity, or “reject”. In this situation, we will be more gentle than a flat rejection, and ask for a wholly new solution, and perhaps provide suggestions of where to begin.

Midterm (30%) and Final (40%)

There will be two exams each quarter, a midterm in week 6 of each quarter and a final exam in week 11 (finals week) of each quarter. We will discuss the format of the exams. In general the level of difficulty will reflect the difficulty of problems on the qualifying exam.

We will cover most of Probability and Stochastics by Erhan Çinlar. I anticipate we will cover the first four chapters in Winter quarter, and the remaining chapters (perhaps focusing on particular applications/examples as we move deeper) in Spring. This schedule may be adjusted as necessary.

Math 672 (Winter)

  • Measure and Integral: $\sigma$-algebras, monotone classes, kernels.
  • Probability spaces: Random variables, expectations, moments, existence and independence.
  • Convergence: Almost sure convergence, convergence in probability, $L^p$ convergence, convergence in distribution. Law of Large Numbers, Central Limits.
  • Conditioning: Conditional expectations, probabilities, distributions and independence. Construction of probability spaces.

Math 673 (Spring)

  • Martingales and Stochastics
  • Poisson Random Measures
  • Levy Processes
  • Brownian Motion
  • Markov Processes

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