Probability Theory II

Syllabus

• Kolmogorov's axioms of probability; distributions, distribution functions, density functions, and discrete and absolutely continuous random variables.
• Transformations of densities under diffeomorphisms.
• Independence of events, event systems, random variables, and generated sigma-algebras; Kolmogorov's zero-one law.
• Main modes of convergence for random variables: convergence in probability, almost sure convergence, and convergence in L^p; relations between them.
• Uniform integrability and the de la Vallée Poussin theorem.
• Lévy's inequality; equivalence of convergence in probability and almost sure convergence for sums of independent random variables.
• Weak laws of large numbers, including Feller's weak law.
• Weak convergence of probability distributions; tightness, relative compactness, and Prohorov's theorem.
• Characterisation of convergence in distribution via distribution functions; Helly–Bray selection theorem.
• Characteristic functions, inversion formula, Doob's inequality, and the continuity theorem.
• Central limit theorem via characteristic functions; Lindeberg–Feller theorem and rates of convergence, including the Berry–Esseen theorem.
• Conditional expectation: general definition, basic properties, computation, conditional densities, and regular conditional distributions.
• Martingales, maximal inequalities, and the martingale convergence theorem.
• Strong laws of large numbers; proof techniques using Kronecker's lemma and martingale convergence; Kolmogorov's strong law.
• Convergence of infinite series of independent random variables; Kolmogorov's three-series theorem.
• L¹ convergence of martingales and regular martingales.