Probability II

Prerequisites: Measure and Integration. Probability I – is very useful, but is not strictly essential. 

Infinitely divisible laws. Theory of discrete-time martingales: Doob inequality, optional stopping, unequal crossover and convergence. Markov chains, random walks on countable spaces, transience and recurrence. Birkoff ergodic theorem. Weak convergence in Polish metric spaces and Prohorov’s theorem. Brownian motion and Donsker’s Theorem.  

References:
BILLINGSLEY, P. – Convergence of Probability Measures. New York, J. Wiley, 1968.
CHUNG, K. L. – A Course in Probability Theory, 2nd ed., New York, Academic Press, 1974.
NEVEU, J. – Discrete Parameter Martingales. Oxford, North-Holland, 1975.
SHIRYAYEV, A. N. – Probability, New York, Springer-Verlag, 1984.
VARADHAN, S. R. S. – Probability Theory, New York, Courant Institute of Mathematical Sciences, 2001.

* Standard program. The teacher has the autonomy to make any changes.