The main goal of this third and last part of the course is the study of several advanced topics in the theory of probabilities, such as: the strong law of large numbers via and for martingales; large deviations type estimates; the central limit theorem and all that it entails (i.e. the in-depth study of the convergence in distribution and the characteristic function); the definition and construction of the Brownian motion in dimension one; some important further results such as the functional central limit theorem.

Prerequisites

Measure theory, a more basic probability course, some functional analysis, the first two parts of the course.

Professor

Nome:   Silvius Klein

Email:     silviusk [at] puc-rio [dot] br

References

[Williams]   David Williams, Probability with Martingales.

[Tao]   Terence Tao, Several of his "What's new?" blog entries.

[Morters-Peres]   Peter Mörters and Yuval Peres, Brownian Motion.

[Durrett]   Richard Durrett, Probability: Theory and Examples.

Grades

Homework problems, due on January 24, 2026.

A written exam on TBA

Final grade calculation: 50% homework (from the three parts together) + 50% written exam.

Lecture notes

The following notes were typed by Mohammed El-Mustafa Ahmed. Thank you so much Mustafa for the help. I still plan on edditing them, but unfortunately this may not happen very soon.

Lecture 1 Introduction to the course; review of martingales; stopping times and martingale transform

Lecture 2 The moment method. The law of large numbers for i.i.d. random variables and for martiangales. Bernstein's trick.

Lecture 3 Large deviations type estiamates: Hoeffding inequality and Azuma-Hoeffding inequality. McDiarmid’s Inequality, an application. Statement of the central limit theorem.

Lecture 4 Reveiw of certain basic concepts, including the distribution of a random variable. Convergence in distribution. Portmanteau's theorem.

Lecture 5 The cummulative distribution function and another characterisation of convergence in distribution. Review of the Fourier transform, the Schwartz space. The characteristic function.

Lecture 6 Lévy's continuity theorem. The completion of the proof of the central limit theorem. Brownian motion: motivation, history, formal definition.

Lecture 7 Gaussian random vectors. Beginning of the construction of Brownian motion in dimension one.

Lecture 8 Completion of the construction of Brownian motion in dimension one. Further properties and applications of brownian motion: Holder continuity; nowhere differentiability; martingale property; Skorokhod’s Embedding Theorem; the functional central limit theorem.