# RANDOM VARIABLES AND STOCHASTIC PROCESSES

## Learning outcomes of the course unit

Since lacking a more advanced course in probability, this course gives a quick but quite comprehensive collection of the main themes of modern probability theory, with a particular attention towards martingale theory and Markov chains.

## Prerequisites

- Elementi di probabilità
- Teoria della misura e dell'integrazione

## Course contents summary

Probability spaces. Sigma-fields. Random variables. Induced law. Absolute continuity. Properties of cumulative function. Measurability lemma. Skorokhod theorem. Mathematical expectation. Monotone convergence. Fatou lemmas, dominated and Scheffé convergence. Jensen inequality. L^p norm. Borel-Cantelli lemmas. Inclusion-exclusion principle.

Markov inequality, Chebyshev inequality, with exponential-optimized version. Legendre transform. Type of convergence for sequences of random variables. Uniform integrability.

Conditional expectation. Existence in L^2 (by projection) and in L^1 (by density). Properties.

Independence of sigma-fileds. Strong law of large numbers (through Garsia lemma, uniform integrability, Kolmogorov's 0-1 law).

Weak convergence. Characteristic function. Lévy theorem. Thightness. Central limit theorem.

Martingales. Discrete stochastic integral. Stopping times. Stopped processes. Optional stopping theorem. Martingale convergence theorem. Closed and closable martingales. Lévy downward theorem and Law of Large Numbers (again).

Simple simmetric random walk. First and last time in 0. Reflection principle.

Radon-Nikodym theorem. Hewitt-Savage theorem.

L^2 martingales, Doob decomposition, quadratic variation. Convergence theorem for L^2 martingales.

Branching processes. Extintion probability. Exponential scaling.

Polya urn and urn processes. Some a.s. convergence results.

Markov chains. Strong and weak Markov property. State classification. Invariant measures.

Large deviations (shortly).

## Recommended readings

- D. Williams - Probability with Martingales - Cambridge University Press - 1991
- Z. Brzezniak, T. Zastawniak - Basic Stochastic Processes - Springer 1999
- J. Jacod, P. Protter - Probability Essentials - Springer 1999

## Teaching methods

This course was held *live* in 2007, when it was recorded (audio/video). For subsequent years we make available the recordings for a remote teaching. The teacher receives students to answer questions and clarify doubts.

The exam requires to solve a small number of difficult research-like problems, for which the students have one week. These problems are then discussed during an oral interview.