Catalog description: selected topics in probability such as sums of independent random variables, notions of convergence, characteristic functions, Central Limit Theorem, random walk, conditioning and martingales, Markov chains, and Brownian motion. 

But we do not necessarily follow it! 

Topics:  At the beginning, we'll follow 

• Probability: Theory and Examples (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 49) 5th Edition by Rick Durrett (PDF of Version 5, January 11, 2019, is available HERE ).

Later on, we are going to concentrate on more specific topics. What I have in mind is the modern theory of Markov chains, using the texts

• Markov Chains and Mixing Times 2nd Revised edition by David A. Levin , Yuval Peres (PDF available too here: https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf.),

• My upcoming book (PDF available) with S. Volkov on certain inhomogeneous Markov chains and related random walks.

However, topics will be open to discussion with the students. And we may even switch the order of the above two parts, if starting with elementary Markov chains, because it does not really require any background.

Other background; recommended books to strengthen your background in (measure theoretical) probability: 

• "Measure Theory and Probability Theory" (Springer Texts in Statistics) by  Athreya and  Lahiri

• "Probability And Measure" by  Billingsley

However, Markov chains would require almost no measure theory!

Grading: If you take the course for pass/fail, then you should simply give a presentation. For the others, grades are based on homework.

Prerequisites: MATH 6310 or equivalent. Undergraduates must have approval of the instructor. Restricted to graduate students only.

NOTE: Although 6534 and 6550 do not form a sequence officially, in many ways, 6550 can be considered a continuation of this course.