The aim of the course is to teach the most important combinatorics, linear algebra, real function theory, set theory, complex function theory, functional analysis and geometric tools used in modern probability theory.
We will demonstrate their application in probability theory through examples, but the emphasis is on developing the toolbox. Part of the acquired knowledge will be utilized in the MSc training.
Combinatorics: Generating function method. Stirling formula, Euler Gamma function. Topology: Convergence in metric space and topological space. Compactness. Product space, product topology, Tikhonov's theorem. Linear algebra: Inner product spaces, Cauchy-Schwarz inequality. Matrix exponentiation, analytic matrix calculus. (Application: Markov transition probabilities.) Function transforms: Laplace transform. Fourier series expansion, Fourier transform, discrete Fourier transform. (Application: characteristic function.) Legendre transform. Measure theory: Interchangeability of integration and differentiation. Uniform convergence and continuity. (Application: differentiability of characteristic function.) Jensen's inequality. Absolute continuity, Radon-Nikodym theorem. (Application: conditional expectation.) Pushforward of measures, integration with substitution. (Application: Distribution of probability variables, expected value of distributions.) Product space, product measure. Fubini's theorem. (Application: independence.) Decomposition of measures, conditional measure, factor measure. Complex function theory: Residue theorem, Laurent series expansion. (Application: calculation of convolutions and characteristic functions.) Analytic extension, Vitali's theorem. Functional analysis: Spectrum of bounded operators, resolvent, spectral radius. Hahn-Banach theorem. C^k spaces, Arsela-Ascoli theorem. Continuous linear functionals, Riesz-Markov theorem. Dual spaces, weak star topology, tightness. Fourier transform once again, Riesz-Fischer theorem.