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Abhishek Sinha

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, L2 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, Lp 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