MATH 5160-5161  : Probability Theory and Stochastic Processes

Catalog description: Convergence of random variables and their probability laws, maximal inequalities, series of independent random variables and laws of large numbers, central limit theorems, martingales, Brownian motion.

Extended description: Foundation of probability theory, monotone classes and pi-lambda theorem, Kolmogorov extension theorem and infinite product spaces, Kolmogorov zero-one law, a.s. convergence, convergence in probability and in Lp of random variables, Borell-Cantelli lemma. Convergence of series of independent random variables: the theorems of Kolmogorov and Levy. Weak convergence of probability measures: characteristic functions, Levy-Cramer continuity theorem, tightness and Prohorov's theorem. The Central Limit Theorem: the Lindeberg-Feller theorem, the Levy-Khintchine formula, stable laws. Conditional expectation. Discrete time (sub- and super) martingales: Doob's maximal inequality, Optional Stopping Theorem, uniform integrability, and the a.s. convergence theorem for L1 bounded martingales, convergence in Lp. Definition, existence and basic properties of the Brownian Motion. Other topics in probability theory at the choice of the instructor (e.g. Markov chains, Birkhoff-Khinchine and Kigman ergodic theorems, Levy's arcsine law, Law of Iterated Logarithm, convergence to stable laws). See also Probability Prelim Study Guide. Textbook choices: Real Analysis and Probability by R.M. Dudley. Probability, Theory and Examples, 4th edition by R. Durrett. Probability Theory by S.R.S. Varadhan. Foundations of Modern Probability by O. Kallenberg. Supplementary reading: Lecture Notes by R. Bass. A Course in Probability by K.L. Chung. Probability with Martingales by D. Williams.
Prerequisites: MATH 5111
Credits: 3