List of topics covered in Stat 330/600 in spring 2017

  1. Why bother with measure theory? Advantages and disadvantages of countable additivity. Sigma-fields are forced upon us. Why Lebesgue was a genius.
  2. Generating classes for sigma-fields. The importance of picking the generating class to suit the problem.
  3. Measurable functions. The benefits and disadvantages of allowing a real function to take infinite values.
  4. The cone ℳ+ of [0,∞]-valued measurable functions. Approximation by simple functions.
  5. Integrals as "increasing `linear' functionals on ℳ+ with the monotone convergence property". ℒ1 as a vector space.
  6. Image measures. Distributions of random variables, random vectors, and stochastic processes.
  7. Dominated Convergence. Differentiation under integral signs.
  8. Negligible sets. ℒ1 versus L1.
  9. Convexity-based concepts: ℒp and Orlicz spaces; Hölder inequality; Jensen's inequality; subgaussianity. The clever converse to Borel-Cantelli.
  10. The π–λ theorem for sets.
  11. Independence of sigma-fields.
  12. Strong law of large numbers with fourth moments. (Better versions later.)
  13. Independence and product measures. Lambda spaces of functions (important: replaces lambda cones in UGMTP). The π–λ theorem for functions. Tonelli and Fubini. Convolutions.
  14. Quantile transformations.
  15. Construction of measures on product spaces via kernels. Bayes. Brief mention of disintegration.
  16. Conditional probability distributions.
  17. Projections in ℒ2. (Hilbert spaces.) Radon-Nikodym. Kolmogorov conditional expectations.
  18. Measure preserving transformations and the Ergodic theorem. Comment on SLLN.
  19. Martingale et al. Stopping times. Information available at a stopping time. The Stopping Time Lemma (a.k.a. the magic lemma). Using STL to prove maximal inequalities. Convergence of positive supermartingales via Dubins's inequality. Applications.
  20. Convergence in distribution for metric spaces. Bounded Lipschitz functions. Continuous Mapping Theorem.
  21. Central limit theorem. Lindeberg and beyond.
  22. Stochastic order symbols.
  23. This year the material on Fourier transforms (a.k.a. characteristic functions) fell off the end of the course.