PROBABILITY Qualifying Examination Syllabus

1. Concepts of Probability: Combinatorial analysis, conditional probability, random variables and their distributions, generating functions, sums of independent random variables.
2. Measure Theoretic Foundations: Probability measures, random variables, transformations of random variables, distribution functions, expectation, modes of convergence of random variables, Kolmogoroff existence theorem, independence, conditional expectation.
3. Law of Large Numbers: Weak law of large numbers, Borel-Cantelli lemma, zero-one laws, Kolmogoroff maximal inequality, 3-series theorem, strong law of large numbers.
4. Convergence in Distribution: Weak convergence of probability measures, Portmanteau theorem, characteristic functions, Levy continuity theorem, central limit theorems including Lineberg-Feller central limit theorem, infinite divisibility.
5. Stochastic Processes: Random walk, Poisson process, Markov chains, martingales and Doob's martingale convergence theorem.

Suggested Textbooks

• Ash, R.: Real Analysis and Probability
• Billingsley, P.: Probability and Measure - All sections except 9,15-19,30-32,38
• Chow and Teicher: Probability Theory
• Chung, K.: A Course in Probability Theory - All sections except 7.5, 8.4, 8.5
• Dudley, R.: Real Analysis and Probability
• Durrett, R.: Probability: Theory and Examples (Especially sections 1.1-1.8, 2.1-2.6,3.1,3.2,4.1-4.4,7.1)
• Feller, W.: An Introduction to Probability Theory and Applications
• Grimmet and Welsh: Probability: An Introduction
• Karlin and Taylor: A First Course in Stochastic Processes
• Lamperti, J.: Probability
• Ross, S.: A First Course in Probability
• Ross, S.: Introduction in Probability Models