Università Roma Tre

Branching processes, introduction to Sigma-algebras, measure spaces and probability spaces. Contruction of Lebesgue measure. Pi-systems, Dynkin's lemma. Properties of measures, sup and inf limits of events, measurable functions and random variables. Borel-Cantelli lemmas. Law and distribution of a random variable. Concept of independence. Convergence in probability and almost sure convergence. Skorokhod's representation theorem. Kolmogorov's 0-1 law.
Integrals, their properties and related theorems. Expectation of random variables. Markov, Jensen and Hoelder's inequalities. L^p spaces. Weierstrass' Theorem. Product measures, Fubini's theorem and joint laws.
Conditional expectation and its properties. Martingales, predictable processes. Stopping times and stopped processes. Optional stopping theorem, applications to random walks. Theorems about convergence of martingales. Strong law of large numbers.
Doob's inequalities for martingales and sub-martingales, applications. Characteristic functions and inversion theorem. Fourier transform in L^1. Equivalence between convergence in distribution and convergence of characteristic functions. Central limit theorem.

D. Williams, Probability with martingales
R. Durrett, Probability: Theory and examples