# Measure Theoretic Probability (MAE717)

### General

School School of Science Department of Mathematics Undergraduate ΜΑΕ717 5 Measure Theoretic Probability Lectures (Weekly Teaching Hours: 3, Credits: 6) Special background - Greek Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes The object of Probability Theory is the study natural phenomena that are subject to randomness. The aim of this course is the introduce the students to the axiomatic development of probability theory of Kolmogorov in the context of measure theory and the rigorous proof of the central theorems of probability theory. After the end of the course the students will know: The axiomatic development of Probability Theory. The notion of stochastic independence of $$\sigma$$-algebras. The proof of the Law of Large numbers for square-integrable independent and identically distributed random variables. The notion of conditional expectation and martingales. Characteristic functions, weak convergence and the proof of the Central Limit Theorem. Working independently. Working in groups. Creative, analytical and inductive thinking.

### Syllabus

Foundations of Probability theory: Probability Spaces, random variables and measurability, Borel σ-algebras, distribution of random variables, the π-λ theorem of Dynkin and equality of measures. Mean Value: Mean value as Lebesgue integral, $$L^p$$-spaces, image-measure, integration with respect to image measures, density functions as Radon-Nikodym derivatives, distribution functions. Markov-Chebyshev inequality, Jensen inequality. Moment generating functions, Chernoff bounds. Convergence in probability, pointwise convergence of random variables and convergence theorems. Stochastic independence 1: Stochastic independence of sets, σ-algebras and random variables. Criterions of independence. Independence and expectation, convolution and sums of independent variables. Borel-Cantelli lemmas and 0-1 laws of Kolomogorov. Laws of Large numbers: proof of the weak law of large numbers, proof of the strong law of large numbers. Weak convergence of random variables and empirical law of large numbers. Stochastic independence 2: Infinite product of probability spaces, construction of independent and identically distributed sequences of random variables with a given distribution. Proof of the interpretation of probability as relative frequency via the law of large numbers. Conditional expectation: Existence of Radon-Nikodym derivative, existence as projection, main properties and convergence theorems for conditional expectation, the disintegration theorem. Martingales}): Filtrations, adapted sequences, definition of martingales and examples, stopping times, Doob's optional stopping theorem, the convergence theorem. Square integrable martingales, proof of the law of large numbers via martingales. Central Limit Theorem: Characteristic functions, Levy's convergence theorem for weak convergence , proof of the central limit theorem.

### Teaching and Learning Methods - Evaluation

Delivery

Face-to-face

Use of Information and Communications Technology
Teaching Methods
Lectures (13x3) 39
Individual study 78
Exercises/projects 33
Course total 150
Student Performance Evaluation

Greek or English
Weekly exercises, midterm exam, final written exam.

### Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• David Williams, Probability with Martingales of Cambridge Mathematical Textbooks, Cambridge University Press, 1990.
• S.R.S. Varadhan, Probability Theory volume 7 of Courant Lecture Notes in Mathematics, American Mathematical Society, 2001.
• R.M. Dudley, Real Analysis and Probability volume 74 of Cambridge studies in advanced mathematics, Cambridge University Press, 2002.
• Heinz Bauer, Probability Theory and Elements of Measure Theory, 2nd edition, Probability and Mathematical Statistics, Academic Press, 1997.
• Heinz Bauer, Probability Theory, Philosophie Und Wissenschaft (de Gruyter Studies in Mathematics), 1996.
• B. Fristedt and L. Gray, A Modern Approach to Probability Theory, Probability and Its Applications, Birkhauser, 1997.