Tools of Modern Probability -- fall semester 2016
A modern valószínűségszámítás eszközei -- 2016 őszi félév
freely choosable this semester -- ebben a félévben szabadon választható


Subject code: BMETE95AM33
Lectures: Monday 14:15-16:00, room T604 and Wednesday 14:15-16:00, room H405/A
Lecturer: Imre Péter Tóth

Suggested literature
Most of the material deiscussed in class is covered in the following literature. From the books, you only need a tiny bit, detailed below. Unfortunately, part of what I said is not covered in any of these, and I couldn't find good literature (which is not unresonably long and hard for the little that I said). For other parts, I do suggest something, but it fits only partially. These will be indicated.

• [TIP] Draft lecture notes written by the lecturer: Tools of Modern Probability.
• [D] R. Durrett: Probability. Theory and Examples. 4th edition, Cambridge University Press, 2010.
• [R_RCA] W. Rudin: Real and complex analysis. 3rd edition, McGraw-Hill Book Company, 1987.
• [R_FAG] W. Rudin: Fourier analysis on groups. Wiley Classics Library Edition, 1990.
• [no] No good suggestion

What actually was covered: detailed list with references

• o and O-notation, asymptotic equivalence [no]
• Gaussian integrals [TIP, section 1]
• Polar coordinates in higher dimensions, surface of hyperspheres [TIP, section 2]
• Almost Gaussian integrals, Laplace's method [TIP, section 3]
• Euler gamma function, Stirling's approximation [TIP, section 4-5]
• Application: de Moivre-Laplace central limit theorem (CLT) [D, section 3.1]
• Measure space, probability space. Push-forward of measures. Distribution of random variables [TIP, section 6.1, 6.2; D, section 1.1-1.5]
• Integral, expectation. Integration by substitution. Expectation of random variables. Densities of measures. Sums of series and Riemannian integrals as special cases of the (Lebesgue) integral. [TIP, section 6.3.1; D, section 1.6]
• Charactersitic functions of random variables, characteristic functions of probability distributions. Characteristic functions and sums of independent random variables. [D, section 3.3.1]
• Exchanging the integral and the limit: monotone convergence theorem, dominated convergence theorem, Fatou lemma. [TIP, section 6.3.2; D, section 1.6.2]
• Application: differentiability of the characteristic function. [D, Exercise 3.3.14]
• Product space, product measure. Exchanging integrals: Fubini's theorem [D, section 1.7]
• Absoulte continuity. Radon-Nikodym theorem (without detailed proof). [TIP, Def. 6.39, Thm 6.40; D, Thm A.4.6]
• Composition of measures and kernels. Decomposition of measures, conditional measure, factor measure. [TIP, section 7]
• Conditional expectation of random variables. Existence, uniqueness. [D, section 5.1]
• Jensen's inequality for conditional expectations. [D, Thm 5.1.3]
• Linear algebra: Diagonalization and spectral decomposition of matrices. Application in calculating powers of matrices. Example: closed formula for the Fibonacci sequence. [no]
• Laplace operator, Laplace equation, harmonic functions. Relation to complex differentiability in 2 dimensions. Conformal maps, conformal equivalence of domains. Riemann mapping theorem. Linear fractional transformations. Applications in solving the Laplace equation. [no]
• Discrete Laplace transform, hitting probabilities of random walks and the Wiener process. [no]
• Fourier transform of L^1 functions. Exiestence, Riemann-Lebesgue lemma. Inverse Fourier transform [partially: R_RCA, section 9.1, 9.2] When is the Fourier transform in L^1? [D, Thm 3.3.5]
• Circle, torus and periodic boundary conditions. Characters on the line and on the torus. Variants of Fourier transformation: Fourier expansion, discrete time Fourier transform, discrete Fourier tranform. [partially: R_FAG, section 1.2.7]
• Weak convergence of random variables, weak convergence of probability distributions. Construction of random variables with a given distribution. The relation of weak convergence and strong convergence. Equivalent formulations. [D, Thm 3.2.2, Thm 3.2.3]
• Existence of weak limits -- vague convergence, Cantor's diagonal argument. Tightness and weak convergence. [D, Thm 3.2.6, Thm 3.2.7]
• Application: continuity theorem for characteristic functions. [D, section 3.3.2]

A few more topics, which were discussed, but will not be asked on the exam:

• Compactness in topological spaces, sequential compactness in metric spaces.
• Products of metric spaces and topological spaces. Tychonoff's theorem, sequential Tychonoff theorem.
• Normed spaces and their duals. Local compactness: Banach-Alaoglu theorem, sequential Banach-Alaoglu theorem.

Grading rules:

• There are/will be homeworks. These may be handed in, but are not compusory. If handed in, they will be corrected and given back.
  
• The grade will be given based on an oral exam. On this exam, I will give at least one practice exercise, and at least one of the homeworks. This practical part contributes to the grade with 30% weight.
  
• Since this is an experimental course, and much of the material is badly documented, each student can choose their favourite half of the material, from which I ask.