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 14:1516:00, room T604 and Wednesday 14:1516:00, 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, McGrawHill 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 Onotation, 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 45]

Application: de MoivreLaplace central limit theorem (CLT) [D, section 3.1]

Measure space, probability space. Pushforward of measures. Distribution of random variables [TIP, section 6.1, 6.2; D, section 1.11.5]

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. RadonNikodym 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, RiemannLebesgue 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: BanachAlaoglu theorem, sequential BanachAlaoglu 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.