M16 Course description
(content subject to revision as the course evolves)
This course is intended to cover some of the basic probability and Markov
chains which is used in other M.Phil courses. Much of the material is covered
in some form in Cambridge undergraduate courses. The course will consist
of the extensive study of some fundamental examples rather than a formal
exposition of the theory. This course assumes only a good general background
in mathematics, including an introductory course in probability. The following topics will be discussed:

Basic tools. Probability spaces, random variables. Distribution,
discrete r.v.s, absolutely continuous r.v.s. Expectation, variance, Markov
inequality, Chebyshev inequality. Independence. Joint distribution, conditional
probability, conditional expectation. Characteristic functions.

Fundamental probability results. Laws of large numbers. Convergence
of random variables, convergence of distributions. Central limit theorem.

Markov chains. Definition, examples. Classification of states, irreducibility.
Recurrence and transience. Stationary distributions and longterm behaviour.
Continuous time Markov processes. Poisson process, examples of queues.

Martingales. Definition, examples. Optional stopping, L^{2}
convergence theorem.
Books:

Williams, D., Probability with Martingales, Cambridge UP (1991)

Norris, J. R., Markov Chains, Cambridge UP (1997)

Grimmett, G. R. and Stirzaker D. R., Probability and Random Processes,
Clarendon (1992)

Durrett R., Probability: Theory and Examples, Duxbury (1991)

Ross, S. A first
course in Probability Theory, edited by Pearson.