## 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

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