## Math 536 (Probability Theory). Spring 2023

Instructor: Steve Krone

Office Hours: by appointment.

Class Time: MWF 12:30 - 1:20
Place: TLC 145

Prerequisite: Math 535 (measure theory and integration)

recommended Text: R. Durrett, Probability: Theory and Examples, Cambridge University Press

This is a first graduate course in probability, based on measure theory. The goal is to learn the mathematics needed to model and analyse random phenomena. Math 451 (undergraduate probability) is not a prerequisite for the course, but having it will certainly help your intuition.

Probability theory is one of the cornerstones of applied mathematics, making rigorous how one models phenomena that cannot be predicted with certainty (basically everything!), when deterministic models are relevant, what kinds of questions can be asked about models with randomness and how one goes about doing the analysis to answer them. Moreover, and just as importantly, probability theory has deep connections with other parts of mathematics, including partial differential equations, real and functional analysis, and ergodic theory. For example, Brownian motion leads to insights about the heat equation, and vice versa--sometimes in striking and unexpected ways.

Grades will be based on homework (30%), a midterm exam (30%), and a final exam (40%).

Course Outline:

1. Probability spaces and random variables, expected value, independence, Borel-Cantelli lemma, almost sure convergence, convergence in probability, weak and stong laws of large numbers.
2. Weak convergence, central limit theorems, characteristic functions.
3. Conditional expectation, filtrations, martingales, stopping times, martingale convergence theorem, optional stopping theorem, Doob's maximal inequality, uniform integrability, 0-1 laws.
4. Topics from stochastic processes.