Limit Theorems and Convergence

This course will begin with a number of classical limit theorems such as central limit theorems of sequences or random variable and Martingales and Poison limit theorems. Limit theorems yielding less standard distributions will also be discussed. It will also discuss convergence and limit theorems for Markov chains. At the end we will discuss a number of special topics such as averaging theorems for Markov chains, especially those with different time scales. The emphasis will be on how one proves limit theorems, both standard and nonstandard. At the start of the course we will develop the needed tools.

Background in probability and analysis. This course will assume a basic grounding in graduate probability. Students should have some exposure to basic real analysis. The basic graduate probability class in the statistics department or some exposure to probability at a graduate level and an introductory real analysis course should be sufficient. If you are unsure, please ask the instructor.