Probability theory is now applied in a variety of fields including physics, engineering, biology, economics, social sciences, ... This course is an introduction to the rigorous theory of probability. The perspective theme is the Doob's theory of discrete time martingales. The Kolmogorov strong law of large numbers and the theorem of three series are proved with martingale techniques. In addition, the central limit theorem is proved together with the main results on week convergence and characteristic functions. In the part of exercises, the first results of Markov chains are introduced.
Expected learning outcomes
Knowledge of the topics of the course and their application to theoretical problems.
Part I. Foundations: Measure spaces Events Random variables Independence Integration Expectation Strong law of large numbers Product measure Gaussian vectors
Part II. Martingale Theory: Conditional expectation Martingales The convergence theorem UI Maringales, L1 convergence and applications L2 Maringales, angle-brackets process and relation with martingale' convergence
Part III. Compendia of theory Markov chains Weak convergence. Tightness. Lévy's Convergence Theorem. Central Limit Theorem