Be aware of the content and meaning of the basic theoretical results related to sums of random variables and discrete Markov chains and their asymptotic properties. Have understood the concepts of transition matrix, trajectory law, state classification, and invariant measures. Be able to consciously reproduce the main steps in the proof of the theory's construction. Be able to frame and solve some problems of probabilistic modeling.
Course Prerequisites
Knowledge of the topics covered in the course of Elements of Probability.
Teaching Methods
Theoretical lessons (32 hours) alternated with exercises (24 hours), in which examples will be discussed and some exercises will be solved on the topics covered in the lessons.
Assessment Methods
The exam consists of two parts. The written exam assesses the skills the student has acquired in calculation and problem-solving related to the topics of the course. The exercises will vary in difficulty, aimed at determining the depth of understanding of these skills. If the student scores higher than 18/30 in the written exam, they will be admitted to the oral exam. In the oral part, the focus will primarily be on verifying the level of knowledge of the concepts presented during the course, the clarity with which they are presented, and the student's ability to apply them. The final grade will be determined by considering the overall breadth and depth of learning, as well as the clarity of presentation and the skills demonstrated in solving problems. The grade is decided by the examination board and will result from the comparison, not necessarily reduced to an arithmetic average, of the evaluations of both the written and oral parts.
Texts
1. G. Grimmett and D. Stirzaker (2020) Probability and Random Processes. Oxford University Press 2. R. Durrett (2016) Essentials of Stochastic Processes. Springer
Contents
PART 1: Sums of Random Variables. - Review of the binomial distribution - Normal approximation of Bernoulli-Laplace - Independence of increments - Success times - Barriers and Reflection Principle - Recurrence and first return times - Trajectory laws PART 2: Markov Chains. - Markov properties, transition matrix - Examples of chains - Trajectory law, Chapman-Kolmogorov, existence hints (Daniell's Theorem) - State classification - Transience, positive and null recurrence - Periodicity - Invariant measures, existence and uniqueness, time reversibility PART 3: Asymptotic Properties. - Review of convergences and classical limit theorems - Ergodic theorem for chains - Convergence to equilibrium - Monte Carlo method
