Probability Theory (7 ECTS)

Non-countable sets impose the need for mathematical foundation of the concept of measurability restricted to appropriate subsets called events, restricting also the definition of any probability measure solely to such measurable/event subsets, thus shaping the constrained concept of probability space (σ-algebra of events abiding by σ-additivity of the probability measure) as Kolmogorov’s basic axiomatic framework for the development of contemporary probability theory on non-countable sets, in unity with classical theory on countable or finite sets where all subsets are considered events. Classes of subsets and the concept of the smallest/least class with specified structure (e.g. π-system, semi-algebra, algebra, monotone class, λ-system, σ-algebra) generated from a given arbitrary non-empty class of subsets, Borel σ-algebras generated from classes of intervals or rectangular subsets in finite dimensional Euclidean spaces. General properties of probability measures, equivalence of σ-additivity and continuity. Extension of a probability measure defined initially on an algebra (or a semi-algebra) of subsets, and uniqueness of its extension to the (complete) σ-algebra of subsets measurable in the Caratheodory-Lebesgue sense with respect to exterior measure induced (on the powerset) by the initial probability measure. Applications of the extension theorem in constructing probability measures on (Euclidean) Borel spaces from a given probability distribution function (univariate on intervals or multivariate on rectangles), on spaces of functions from a given (Kolmogorov consistent) family of finite dimensional distributions on the so-called cylinder-like subsets of functions, and on product probability spaces. Decomposition of a probability measure to a (unique) convex linear combination of potentially three component distributions (discrete, absolutely continuous, singular continuous) with respect to Lebesgue measure on the same (Euclidean) Borel space, connections with the theory of probability distribution functions, Radon-Nikodym characterization of the absolutely continuous distributions as dominated by Lebesgue measure, Cantor’s distribution counterexample and other examples.

Random variables defined as measurable real-valued functions, pointwise convergence of a sequence of random variables defined in the same space, measurability of the limit and of the domain of convergence, Borel measurability and other properties. Stochastic independence among classes of events or random variables, Borel-Cantelli lemmas, terminal (or tail) σ-algebra, Kolmogorov’s 0-1 law. Expectation of a random variable constructed as Lebesgue integral with respect to a probability measure in the space where the random variable is defined and equivalently as Lebesgue integral with respect to the probability distribution (or law) induced by the random variable on the Borel real-line, integrability of a random variable, properties of expected values. Modes of stochastic convergence for sequences of random variables (almost surely, in mean of p-th order, in probability, in law or weakly), their inter-relationships and equivalent characterizations of the two weaker modes, limit theorems (weak and strong laws of large numbers, central limit theorems), expectation continuity theorems (monotone convergence, Fatou’s lemma, dominated or bounded convergence, uniform integrability convergence), characteristic functions, Fourier inversion theorem, method of moments. Conditional expectation of an integrable random variable with respect to a given sub-σ-algebra of events of the probability space and proof of existence of almost surely equivalent versions of it as Radon-Nikodym derivatives of suitable finite measures dominated by the probability measure on the conditioning sub-σ-algebra, properties and examples of conditioning, conditional probabilities of events as conditional expectations of indicator functions corresponding to the same events.

Recommended Bibliography:

• Rosenthal, J. S. (2006): A First Look at Rigorous Probability Theory, 2nd Edition, World Scientific.
• Ρούσσας, Γ. Γ. (1992): Θεωρία Πιθανοτήτων, Eκδόσεις ΖΗΤΗ, Θεσσαλονίκη (11057-ΕΥΔΟΞΟΣ).
• Καλπαζίδου, Σ. (2002): Στοιχεία Μετροθεωρίας Πιθανοτήτων, Eκδόσεις ΖΗΤΗ, (11379-ΕΥΔΟΞΟΣ).

Recommended Supplementary Bibliography:

• Shiryaev, A.N.: Probability, 3rd Edition, (75491026 – Vol. Ι, 2016 & 91693435-Vol. ΙΙ, 2019), Problems in Probability (73251304-ΕΥΔΟΞΟΣ, 2012), Springer-Verlag.
• Billingsley, P. (1995): Probability and Measure, 3rd Edition, John Wiley & Sons, New York.
• Bhattacharya, R. and E.C. Waymire (2016): A Basic Course in Probability Theory, 2nd Edition, Springer (75480799-ΕΥΔΟΞΟΣ, Αρχική έκδοση 2007: 176605-ΕΥΔΟΞΟΣ).
• Roussas, G.G. (2005): An Introduction to Measure-Theoretic Probability, Elsevier-Academic Press.
• Leadbetter, R, S. Cambanis and V. Pipiras (2014): A Basic Course in Measure and Probability – Theory for Applications, Cambrige University Press.
• Chung, K.-L. (1974): A Course in Probability Theory, Academic Press, San Diego.
• Durrett, R. (1996): Probability: Τheory and Εxamples, Duxbury, Belmont.
• Port, S.C. (1994): Theoretical Probability for Applications, John Wiley & Sons, New York.
• Capinski, M. and Kopp P.E. (2004): Measure, Integral, and Probability, 2nd Edition, Springer.
• Athreya, K.B. and S.N. Lahiri (2006): Measure Theory and Probability Theory, Springer (173065-ΕΥΔΟΞΟΣ).
• Skorokhod, A.V. (2005): Basic Principles and Applications of Probability Theory, Springer (172343).
• Gut, A. (2005): Probability: A Graduate Course, Springer (169327-ΕΥΔΟΞΟΣ).