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March 15th, 2012, 08:07 AM   #1
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Summer school course in advanced prob. theory

Dear all,

I would like to take a course in advanced probability theory during the summer. I have searched for a University that offers courses in advanced probability theory but without luck... If any of you have any suggestions please let me know

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March 15th, 2012, 09:44 AM   #2
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Re: Summer school course in advanced prob. theory

What do you mean by advanced probability theory? Are we talking sigma-algebras and Kuratowski theorems, or more like kurtosis, cross-correlations, and convolutions? The first is what I think of as advanced probability, but perhaps you mean more of the second (advanced topics in probability, with some statistics, but less abstract the the former).
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March 15th, 2012, 02:33 PM   #3
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Re: Summer school course in advanced prob. theory

You are right. I should be more specific! I look for courses that cover some (not all) of the following topics:

1) Sigma-algebras: definition, properties, Borel-sigmaalgebra.
2) Measures: definition, properties, Lebesgue measure, probability measures.
3) Uniqueness theorem (without proof). Distribution functions.
4) Measurable functions: definition, properties, case-by-case result, stochastic variables.
5) Integral: construction, theorems on monotone and majorized convergence, relation to simpler concepts of the integral.
6) Functions, given as integrals: continuity, differentiability, interchanging of limit operations.
7) The function spaces L1 and L2: completeness. Mean and variance.
8) Measures with density. Probability densities.
9) Image measures. Transformation of probability measures.
10) The transformation theorem in 1 dimension.
11) Product measures: definition, the theorems of Tonelle and Fubini, independent stochastic variables.
12) Fouriertransformation (L1).
13) Sequence of random variables, almost sure convergence, Kolmogorov's 0-1 law.
14) The strong law of large numbers.
15) Weak convergence of probability measures.
16) The central limit theorem.
17) The ergodic theorem.

Other topics from measure theory or stochastic processes (Brownian motion) would be nice too... I know it's a lot but I'm looking for a course that approximates a course covering the first 12 points. I need it to take some finance courses so all topics might not be neccesary.
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