Ph.D. Comprehensive Examination Topics
Probability
2005
Measure Theory
Theoretical foundations of measure theoretic probability, random variables, etc.
Convergence
Types of convergence and their relations
Laws of Large Numbers
Chebyshev's Theorem, weak laws, Borel-Cantelli lemmas, methods of proving almost sure convergence, convergence of random series, strong laws, laws under minimal hypotheses
Central Limit Theorems
Central limit theorem (DeMoivre-Laplace Theorem), Stirling's formula, weak convergence, Poisson convergence, characteristic functions
Discrete Probability and Combinatorics
Counting techniques, multiplication principle (Fundamental Principle of Counting), permutations, combinations, distinguishable and indistinguishable objects, with and without replacement, urn models, basic probability distributions, conditional probability, generating functions
Stochastic Processes
random walks, Markov chains, queuing processes, birth and death processes, branching processes
Recommended Texts:
An Introduction to Probability Theory and its Applications, volume I, 3rd edition, by William Feller
Probability: Theory and Examples, 2nd edition, by Richard Durrett
Probability and Measure, 3rd edition, by Patrick Billingsley
A First Course in Stochastic Models, by Henk Tijms
Elements of Applied Stochastic Processes, by U.N. Bhat and G.K. Miller

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