Advanced Probability

Probability spaces. Random variables. Independence. Borel-Cantelli Lemmas. Weak and strong law of large numbers. Iterated Logarithm Law. Birkhoff's Ergodic Theorem. Convergence in distribution. Continuous mapping theorem. Characteristic function. Central Lindenberg-Feller Limit Theorem. Stable distributions. Conditional hope. Conditional probability.

#### Basic Information

Workload

45 hours

Requirements

Measure and Integration

**Mandatory:**

- Athreya, K. B., & Lahiri, S. N. (2006). Measure theory and probability theory. Springer Science & Business Media.
- Rosenthal, J. S. (2006). A first look at rigorous probability theory. World Scientific Publishing Company.
- Williams, D. (1991). Probability with martingales. Cambridge university press.

**Complementary:**

- Durrett, R. (2019). Probability: theory and examples (Vol. 49). Cambridge university press.
- Billingsley, P. (2008). Probability and measure. John Wiley & Sons.
- Bartle, R. G. (2014). The elements of integration and Lebesgue measure. John Wiley & Sons.
- Pedro J. Fernandez. Medida e Integração. Coleção Euclides, IMPA.