B018779 - PROBABILITY

Law of large numbers, Central limit theorem.Large deviation theory. Rigorous probability theory. Weak convergence. Caratteristic function, Decomposition of probability laws. Conditional Probability. Martingales. Stochastic processes: Poisson, gambling games, Markov chains, Random walk, Branching proceses, Brownian motion.

1) Rosenthal, A first look at rigorous probability theory.
2) Frank den Hollander, Large deviations, Chapters 1 e 2.
3) Caravenna Dai Pra, Probabilita`.
4) Baldi, Calcolo delle Probabilità. Capitolo 5.
5) Sheldon Ross, Calcolo delle probabilità.
6) van der Hofstad, random graphs

The course aims to provide the students with fundamental knowledge and understanding about rigorous probability theory for general random variables and stochastic processes. One of the aim is to let the students develop technical skills to apply the knowledge and results to model concrete situations that needs general random variables and to compute probabilities and distributions requested in the situations described in the problem.
The course aims to provide the students with fundamental knowledge and understanding about limit theorems such as Strong Law of large numbers, central limit Theorems and large deviation theorems (with their proofs) with particular emphasis to the different hypothesis one could assume and to the different type convergence one could obtain. One of the aim is to let the students develop technical skills needed to perform asymptotic estimates, to solve concrete problems and to estimate probabilities that cannot be exactly computed. Another aim is to let the students develop technical skills needed to compute quantities related to Poisson processes and Markov chains and provides methods and examples for the solution of exercises that require to model concrete situation with those stochastic processes. Special attention will be paid to help the students to develop communication skills needed to explain with rigorous mathematical language the results explained and proved in class and to give justification of the methods used to solve the exercises. The course aims to stimulate students to develop independent and critical thinking to establish the appropriate results that can be used depending on the concrete situation described in the problem.

Differential and integral calculation in one variable for real functions. Basic knowledge of algebra and geometry.
Basic notions of discrete and continuous random variables.

Lectures and discussion and correction of homework

The exam consists of a written and oral examinations per session. The written exam will have open questions of two types. The first type in which the student should state and prove results explained during the lessons, with the aim to verifying the knowledge, the understanding and the quality of the exposition. A second type in which the questions are conceived to assess the ability of the students to apply their skills to problem modelling and solving, and to give the rigorous justification using formule and the appropriate scientific language.
The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. During the esposition of the definitions and results with their proofs the student has to show the degree of comprehension of the theoretical and applied aspects of the topics treated during the course. In the assessment, special attention is paid to communication skills, appropriate use of mathematical language. In particular, the teacher will pose be short questions on the relations between topics and on possible strategies to adapt the proofs to different hypothesis in order to evaluate the autonomy and critical thinking on the course topics.
Additionally, the students will have the opportunity to perform two partial tests that, if both successful, will allow them to access directly to the oral examination.