20410414 - CP410 - Theory of Probability

Foundations of modern probability theory: measure theory, 0/1 laws, independence, conditional expectation with respect to sub sigma algebras, characteristic functions, the central limit theorem, branching processes, discrete parameter martingale theory.
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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.

Core Documentation

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

Reference Bibliography

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

Type of delivery of the course

Preferably in presence


Preferably in presence

Type of evaluation

The exam consists of two separate parts: the written part and the oral part. In the first one, students will be asked to solve exercises based on the techniques that we have seen in class; 2 hours. The oral part will start with a discussion of the (possible) mistakes that the student did in the first part; afterwards students will be asked to present and comment statements proofs of some of the most important results seen during the lectures.