Math 521/522/523: Advanced Probability
Zhen-Qing Chen
Autumn/Winter/Spring 1999-2000, MWF 11:30-12:20
Math 521-2-3 is a one-year graduate course on Probability Theory.
We will use Rick Durrett's Probability: Theory and Examples, Second
Edition as a textbook. The topics will include
Math 521:
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Probability sample spaces, random variables, distribution functions;
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Expectation and moments;
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Independence;
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Weak and strong laws of Large Numbers,
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Weak convergence, characteristic functions;
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Central Limit Theorems;
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Poisson Approximation;
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Infinitely Divisible and Stable Distributions;
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Random Walks, stopping times, transience and recurrence.
Math 522:
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Martingales, almost sure convergence, Lp convergence,
backwards martingales, optional stopping theorems.
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Markov chains: Markov property, recurrence and transience, stationary
distribution, convergence theorem, Gibbs sampler, Metropolis algorithm,
and exact sampling with coupled Markov chains;
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Birkhoff's ergodic theorems.
Math 523:
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Brownian motions: construction and basic properties;
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Skorokhod's theorem on embedding random walk into Brownian motion, Central
limit theorems for martingales and for dependent variables;
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Empirical distributions and Brownian bridge;
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Laws of the iterated logarithm for Brownian motion and random walks with
finite variance;
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Topics on stochastic calculus and its application in mathematical finance.
Prerequisite: a knowledge of measure theory and Lebesgue integration,
such as is covered in the Real Analysis courses Math 524-5-6 or Math 424-5-6.
A brief review of measure theory will be given in Math 521.
Other Reference Books:
P. Billingsley: Probability and Measures. 1979.
Leo Breiman: Probability. 1968.
K.L. Chung: A Course in Probability Theory, second edition.
1974.
William Feller: An Introduction to Probability theory and its Applications.
Vol.
I, third edition (1968); Vol II, second edition (1971).