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Graduate Courses

12 units

Prerequisites: Undergraduate Probability 21-325 and Measure Theory and Integration 21-720 . In particular, the measure-theoretic prerequisites for this course include

  • Carathéodory's Extension Theorem;
  • Classical convergence theorems (Dominated Convergence, Monotone Convergence, Fatou);
  • Modes of convergence: in measure, almost everywhere, in $ L^p$, and their relations to each other;
  • Products of measurable spaces and Fubini-Tonelli theorems;
  • Radon-Nikodym derivative.


  • Probability spaces, random variables, expectation, independence, Borel-Cantelli lemmas.
  • Kernels and product spaces, existence of probability measures on infinite product spaces, Kolmogorov's zero-one law.
  • Weak and strong laws of large numbers, ergodic theorems, stationary sequences.
  • Conditional expectation: characterization, construction and properties. Relation to kernels, conditional distribution, density.
  • Filtration, adapted and predictable processes, martingales, stopping times, upcrossing inequality and martingale convergence theorems, backward martingales, optional stopping, maximal inequalities.
  • Various applications of martingales: branching processes, Polya's urn, generalized Borel-Cantelli, Levy's 0-1 law, martingale method, strong law of large numbers, etc.
  • Weak convergence of probability measures, characteristic functions of random variables, weak convergence in terms of characteristic functions.
  • Central limit theorem, Poisson convergence, Poisson process.
  • Large deviations, rate functions, Cramer's Theorem.