EE 6112: Topics in Random Processes and Concentrations
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In Spring 2021, I am offering 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, Uniform integrability
- 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), Entropy methods
- Logarithmic Sobolev inequality
- Talagrand's inequality
- Dudley's entropy integral, Sudakov's lower bound
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%)
- Scribing (15%)
- Take-home Mid-term (20%)
- Reading/Research project (15%)
Problem Sets
References
We will not follow any one particular source, in general. However, the following references will be useful