Probability Prelim topics

Topics

  • Probability spaces, random variables, expectation, variance, conditional distributions, Kolmogorov's extension theorem

  • Independence, Convergence in probability, almost sure convergence, Kolmogorov's 0-1 law, Weak law of large numbers, Borel-Cantelli Lemmas, Strong law of large numbers

  • Weak convergence, Characteristic functions, Plancherel's theorem, the Central limit theorem

  • Conditional expectation, Filtrations, Martingales, Doob's decomposition, Stopping times, the optional stopping theorem, the Martingale convergence theorem 

Main References

  1. J.F. Le Gall, Measure Theory, Probability, and Stochastic Processes, Graduate Texts in Mathematics, Series Number 295, Springer, 2022.
  2. R. Durrett, Probability: Theory and Examples, Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 49. 2019.
  3. P. Billingsley, Probability and Measure, Wiley Series in Probability and Mathematical Statistics, 3rd Edition, 1995
  4. D. Khoshnevisan, Probability, Graduate Studies in Mathematics, Volume 80, AMS, 2007.