MA 5400 PROBABILITY THEORY
Unit I: Probability spaces; properties of probability; Random variables and their properties; expectation; Kolmogorov's theorem about consistent distributions; Laws of large numbers; de Finetti theorems; 0-1 laws; convergence of random series; Stopping times; Wald's identity; Markov property, another proof of SLLN.
Unit II: Convergence of laws: selection theorem; Characteristic functions; central limit theorem; Multivariate normal distributions and central limit theorem; Lindeberg's central limit theorem; Levy's equivalence theorem; three series theorem; Levy's continuity theorem; Levy's equivalence theorem.
Unit III: Conditional expectation; Martingales, Doob's decomposition; Uniform integrability; Optional stopping; inequalities for Martingales; Convergence of Martingales; Portmanteau theorem; Metrics for convergence of laws; empirical measures Convergence and uniform tightness.
Unit IV: Introduction to Stochastic processes.
1. Dudley, R. M. Real Analysis and Probability. Cambridge, UK: Cambridge University Press, 2002.
1. Feller, William. An Introduction to Probability Theory and its Applications. Vol. I and II. New York, NY: Wiley, 1968-1971.
2. Ledoux, Michel. The Concentration of Measure Phenomenon. Vol. 89, Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, 2001