Università Roma Tre

1. Probability

Introductory example: The branching process.
Introduction to measure theory.
Measure spaces. Events. Uniqueness and extension of measure.
Probability measures. First Borel--Cantelli lemma.
Random variables, distribution function and law. Indipendence.
Second Borel--Cantelli lemma.
0--1 law for independent random variables.

2. Integration, expected value

Introduxction to integration theory. Monotone convergence theorem.
Expectation. Taking the limit under expectation.
Jensen's inequality. L_p norms.
H\"older inequality and Cauchy-Schwarz.
Markov's inequality.
Examples of weak and strong laws of large numbers.
Product measures. Fubini theorem.
Joint laws.

3. Conditional expectation, martingales and limit theorems

Conditional expectation with respect to a sub $\sigma$--algebra. Kolmogorov existence and uniqueness theorem.
Filtrations. Martingale. Gambilng. Stopping times.
Optional stopping.
Some applications to exit times from an interval.
Convergence theorem for martingales in L_1 and in L_2. Kolmogorov's strong law of large numbers.

4. Convergence ind distribution and the central limit theorem

Characteristic functions. Inversion theorem.
Equivalence between convergence in distributiona nd pointwise convergence of characteristic functions.
Various modes of convergence for random variables. Examples.

D. Williams.
Probability with martingales
Cambridge University Press, 1991


R. Durrett
Probability: Theory and Examples
Thomson, 2000