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We will cover the following topics.
1. Preliminary
- Random variable
--- probability space, probability distribution, probability density function, expectation, independence
--- important examples: exponential distribution, Gaussian distribution, Gaussian vectors
- Conditional expectation
--- definition, basic properties
--- computation of conditional expectation
- Convergence
--- strong and weak convergence
--- vague topology, Prokhorov’s theorem
- Basic measure theory
--- absolutely continuity, Radon–Nikod´ym theorem
--- Lebesgue’s decomposition theorem
2. Limit theorems
- law of large numbers
--- topology on infinite product space, i.i.d. sequence
--- strong law for i.i.d. sequence
- central limit theorem (i.i.d. case)
- large deviations (i.i.d. case)
--- moment-generating function, Legendre transformation
3. Stochastic process
- Basic concepts
--- sample paths, distribution, finite dimensional distribution
- Poisson process
--- Poisson distribution, distribution of Poisson process
--- semigroup, infinitesimal generator
--- limit theorems, homogenization
- Brownian motion*
--- construction of Brownian motion*, Donsker’s theorem*, Wiener measure*
--- Brownian sample paths*
--- a first look at (Ito) stochastic integral*
4. Poisson point process
- introduction to point process
--- definition of Poisson process as a point process
- Poisson point process on Rd*
--- definition*, law of large numbers*, central limit theorem*
A draft of lecture notes can be found here.
Course Schedule
Mon Tue Wed Thu Fri
6/11 9-11 7/11 9-11 9/11 9-11
13/11 9-11 14/11 9-11 15/11 9-11
20/11 9-11 21/11 9-11 23/11 9-11