EE 6112: Topics in Random Processes and Concentrations
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In Spring 2019, I am teaching an advanced graduate-level course on Probability Theory. This course will tentatively cover the following topics.
A. Review of Basic Probability Theory
- Measure Spaces, Sigma-Algebras, Random Variables
- Expectation, Convergence Theorems
- Conditional Expectation, Filtrations, $\mathcal{L}^2$ theory
B. Martingales
- Definitions, basic properties, sub-martingales, and super-martingales, examples
- Doob's decomposition, stopping times, gambling strategy, optional stopping theorem
- Martingale convergence theorem, Doob-Kolmogorov's inequality, $L^p$ inequality
- Random Walks, exchangeability, de Finetti’s theorem
- Applications to queueing theory, information theory
C. Large Deviation Theory
- Large Deviation for i.i.d. variables, Chernoff Bound, Legendre transform
- Cramer's theorem, rate function and its properties, change of measures
- Gartner-Ellis theorem, Large Deviation for Markov chains
- Applications to queueing theory, insurance
D. Concentration Inequalities
- Martingale concentrations (Azuma-Hoeffding, Doob’s martingale method, median concentrations), MGF methods
- Logarithmic Sobolev inequality
- Talagrand's inequality
- Hanson-Wright inequality
Evaluations
Problem-solving will be our primary vehicle for learning the material. We will have bi-weekly problem sets, a mid-term and a final project. The final grade will be a weighted average of these three components as detailed below:
- Problem Sets (50%)
- Take-home Mid-term (20%)
- Reading/Research project (30%)
Problem Sets
Mid-Term
References
We will not follow any one particular source, in general. However, the following references will be useful
Acknowledgment
This course was originally designed by Prof. Krishna Jagannathan. Here is the link to the previous (2016) offering of this course.