### MATH 5160: Probability Theory and Stochastic Processes I

**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 *L ^{p}* 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

*L*

^{1}bounded martingales, convergence in

*L*. 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).

^{p}**Prerequisites:** None

**Credits:** 3

**Sections: **Fall 2009 on Storrs Campus

Course | Sec | Comp | Time | Room | Instructor |
---|---|---|---|---|---|

5160 | 001 | Lecture | TuTh 11:00-12:15 | MSB307 | Evarist Gine-Masdeu |