Probability Theory (8 ECTS)

Course Code: 
6116
Semester: 
6th
Elective Courses
Διδάσκων: 

PAVLOPOULOS HARALAMBOS

Uncountable sets and the necessity for axiomatic foundation of probability spaces (σ-algebra of events, Kolmogorov’s axioms, properties of probability measure). The Lebesgue-Caratheodory extension theorem for construction of probability spaces (summary, applications). Definition of random variables and Borel measurability. Stochastic independence, Borel-Cantelli lemmas, tail events and Kolmogorov’s 0-1 law. Expectation of random variables with respect to a probability measure as Lebesgue integral with respect to their probability distributions induced on the Borel line, properties of expected values. Modes of convergence for sequences of random variables (almost certain, in p-th order mean, in probability, in distribution). Limit theorems (monotone convergence, Fatou’s lemma, dominated/bounded convergence theorem, uniform integrability, weak and strong laws of large numbers, central limit theorem). Lebesgue’s decomposition of a probability distribution on the Borel line to its components (discrete, absolutely continuous, singular continuous), characterization of absolute continuity by the Radon-Nikodym theorem. Conditional expectation, conditional probability and their properties.

Recommended Reading:

  • Athreya, Krishna B., Lahiri, Soumendra N., Measure Thery and Probability Thery, Springer Science and Business Media, LLC, 2006.
  • Billingsley, P. (1995): Probability and Measure, 3rd Edition, John Wiley & Sons.
  • Bhattacharya, Rabi. Waymire, Edward C., A Basic Course on Probability Theory, Springer Science and Business Media, Inc., 2007.
  • Rosenthal, J. S. (2006): A First Look at Rigorous Probability Theory, Second Εdition, World Scientific.
  • Roussas, G.G. (2005): An Introduction to Measure-Theoretic Probability, Elsevier Academic Press.
  • Skorokhond, A.V., Prokhorov, Yu.V., Basic Principles and Applications of Probability Theory, Springer-Verlag Berlin Heidelberg, 2005.
  • SpringerLink (Online service), Gut A., Probability: Α graduate Course, Springer Science and Business Media, Inc., 2005.
  • Ρούσσας, Γ. Γ. (1992): Θεωρία Πιθανοτήτων, Eκδόσεις ΖΗΤΗ, Θεσσαλονίκη.
  • Καλπαζίδου, Σ. (2002): Στοιχεία Μετροθεωρίας Πιθανοτήτων, Eκδόσεις ΖΗΤΗ, Θεσσαλονίκη.

The courses outline can be found here.