Empirical process theory (EPT) is said to be the great unifier of modern statistics. An example application is stochastic
optimization where EPT is the natural framework if one wishes to unify different spaces, e.g,
function-space, Euclidean space, space of measures etc. over which optimization is performed. And, since weak convergence principles
form some of the basic building blocks for EPT, a strong foundation in weak convergence is a pre-requisite for understanding EPT. Following this logic, this study group
will focus on the basics of weak convergence of probability measures, covering the first three chapters of the book "Convergence of
Probability Measures, Second Edition" by Billingsley.
| lecture number 1 | |
Wednesday, 09/22, 3:00 pm -- 4:00 pm (Pasupathy) |
| measures and integrals (Thms. 1.1, 1.2) |
| Section 1 |
| lecture number 2 | |
Wednesday, 09/29, 2:30 pm -- 3:30 pm (Pasupathy) |
| separability, completeness, separating classes examples |
| Preliminaries |
| lecture number 3 | |
Monday, 10/04, 3:00 pm -- 4:00 pm (Pasupathy) |
| tightness, Portmanteau theorem (Thms. 1.3,2.1) |
| Section 2 |
| lecture number 4 | |
Wednesday, 10/20, 3:00 pm -- 4:00 pm (Pasupathy) |
| convergence determining class, examples. |
| Section 2 |
| lecture number 5 | |
Wednesday, 10/27, 3:00 pm -- 4:00 pm (Pasupathy) |
| mapping theorem, examples; random elements. |
| Section 2 Section 3 |
| lecture number 6 | |
Wednesday, 11/03, 3:00 pm -- 4:00 pm (Pasupathy) |
| random elements, key theorems. |
| Section 3 Section 3 |
| lecture number 7 | |
Wednesday, 11/10, 3:00 pm -- 4:00 pm (Pasupathy) |
| relative compactness, Prohorov's theorem. |
| Section 5 |
| lecture number 8 | |
Wednesday, 11/17, 3:00 pm -- 4:00 pm (Honnappa) |
| tightness in C[0,1]. |
| Section 5 |
| lecture number 9 | |
Wednesday, 12/01, 3:00 pm -- 4:00 pm (Honnappa) |
| Wiener measure existence; Etemadi's inequality. |
| Section 8, Appendix M19 |
| lecture number 10 | |
Wednesday, 12/08, 3:00 pm -- 4:00 pm (Honnappa) |
| Etemadi's inequality; empirical cdf weak convergence. |
| Appendix M19, Section 14 |