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Graduate Courses
21721
Probability
12 units
Prerequisites: Undergraduate Probability 21325 and Measure Theory and Integration 21720 . In particular, the measuretheoretic prerequisites for this course include
 Carathéodory's Extension Theorem;
 Classical convergence theorems (Dominated Convergence, Monotone Convergence, Fatou);
 Modes of convergence: in measure, almost everywhere, in , and their relations to each other;
 Products of measurable spaces and FubiniTonelli theorems;
 RadonNikodym derivative.
Description
 Probability spaces, random variables, expectation, independence, BorelCantelli lemmas.
 Kernels and product spaces, existence of probability measures on infinite product spaces, Kolmogorov's zeroone law.
 Weak and strong laws of large numbers, ergodic theorems, stationary sequences.
 Conditional expectation: characterization, construction and properties. Relation to kernels, conditional distribution, density.
 Filtration, adapted and predictable processes, martingales, stopping times, upcrossing inequality and martingale convergence theorems, backward martingales, optional stopping, maximal inequalities.
 Various applications of martingales: branching processes, Polya's urn, generalized BorelCantelli, Levy's 01 law, martingale method, strong law of large numbers, etc.
 Weak convergence of probability measures, characteristic functions of random variables, weak convergence in terms of characteristic functions.
 Central limit theorem, Poisson convergence, Poisson process.
 Large deviations, rate functions, Cramer's Theorem.
