Tools of Modern Probability -- fall semester 2022

Subject code: BMETE95AM33

Classes:
• Wednesday 10:15-12:00, room H406
• Thursday 12:15-14:00, room H45/a

Lecturer: Imre Péter Tóth

HOMEWORK assignments:
• HW1: Exercises 1.6, 2.2, 3.3 and 4.1 form the practice exercises. Deadline: 3 November 2022.
• HW2. Exercises 5.2, 5.5 and 5.6 form the practice exercises. Deadline: 17 November 2022.
• HW3. Exercises 6.7, 6.8, 6.9, 10.13 and 10.14 form the practice exercises. Deadline: bring them for the exam, or submit online in advance.
Paractice exercises Here.

Course organization:
• There will be lectures, homeworks and an oral final exam. The homeworks contribute to the grade with 40% weight.
• Detailed lecture notes and lecture videos will be available online, on the course home page.
• The lectures consist of theoretical and practice parts, but these will not be strictly separated in time.
• Over the course, the maximum total score is 100: 40 for the homeworks and 60 for the final oral exam.
• The students get homeworks with a total value of 60 scores, but only the best 2/3 is taken into account.
• On the oral exam, every student will get at least one practice exercise.
• To obtain the signature, at least 50% of the homework scores - which is 20 scores - must be reached.
• To pass the exam, at least 50% of the exam scores - which is 30 scores - must be reached.
• Final score to grade conversion table:  score grade 0-49 1 50-62 2 63-75 3 76-88 4 89-100 5
HOMEWORK SUBMISSION: Home works can be submitted either on paper, or electronically via the Moodle system in pdf. If you submit online, please make sure that the file you upload is a single pdf, well readable and of moderate size. (The limit is 10 MB, but 1 MB is enough for everything.)

Planned topics: for sure, not all of these will be covered.
• Gaussian integrals and a geometric application: surface of hyperspheres. Almost Gaussian integrals, Laplace's method. Euler gamma function, Stirling's formula.
• Measure theoretic foundations of Probability theory: integral and expectation; push-forward of measures and distribution of random variables; integral substitution and expectation of distributions. Density of measures and random variables. Exchangeablilty of expectation and limit: monotone and dominated convergence theorems, Fatou's lemma. Fubini's theorem.
• Weak convergence of random variables and distributions. Construction of a random variable with a given distribution. Relation of strong and weak convergence. Existence of weak limits - Cantor's diagonal argument, tightness: Helly's theorem.
• Hilbert spaces, Riesz representation theorem. L^2 spaces: completeness. Absolute continuity, Radon-Nikodym theorem.
• Fourier series and Fourier transform, point spectrum and continuous spectrum. Spectral theorem and Gelfand's spectral radius formula.
• Conditional expectation of random variables - existence, uniqueness. Jensen's inequality for conditional expectations.
• Basics of normed spaces: spaces of continuous functions; measures as functionals (weak* topology), Arsela-Ascoli theorem. Hölder's inequality, Hahn-Banach theorem, Banach-Alaoglu theorem.
• A few basic partial differential equations: wave equation, heat equation. Laplace operator, harmonic functions. Relation with complex differentiability in two dimensions.
• Conformal mappings, conformal equivalence of domains. Riemann mapping theorem. Application of linear fractional transformations in solving the Laplace equation. Discrete Laplace operator and equation: hitting probabilities for random walks and the Wiener process.
Videos of lectures from 2020 that were live:
These videos are not cut at the natural boundaries of the material presented, but at the brakes of the classes where they were recorded. So their descriptions only describe the content roughly.
Lecture 1, part 1: Gaussian integrals (46:21, 357 MB)
Lecture 1, part 2: Spherically symmetric integrals (32:57, 263 MB)
Lecture 2, part 1: Surface of hyperspheres (45:19, 301 MB)
Lecture 2, part 2: Euler Gamma function (32:41, 134 MB)
Lecture 3, part 1: Notation for asymptotic behaviour of functions SORRY, no video: I used the wrong settings, and the quality is unacceptable.
Lecture 3, part 2: Almost Gaussian integrals 1 (49:15, 226 MB)
Lecture 4, part 1: Almost Gaussian integrals 2 (35:14, 134 MB)
Lecture 4, part 2: liminf/limsup and Almost Gaussian integrals 3 (45:14, 200 MB)
Question session (no official class on sports day): Integration of spherically symmetric functions repeated (1:02:00, 323 MB)
Lecture 5, part 1: Examples of almost Gaussian integrals (52:40, 368 MB)
Lecture 5, part 2: Normalizing almost Gaussian functions to Gaussian; Stirling's approximation (27:18, 164 MB)
Lecture 6, part 1: Beta function (36:40, 138 MB)
Lecture 6, part 2: Gamma and beta distributions in Statistical Physics (43:34, 164 MB)
Lecture 7, part 1: Measure theory basics 1 (44:00, 145 MB)
Lecture 7, part 2: Measure theory basics 2 (36:36, 117 MB)
Lecture 8, part 1: Sorry, no video - I managed to record the break instead.
Lecture 8, part 2: Distributions of random variables (33:33, 92 MB)
Lecture 9, part 1: Remarks on homework solutions (20:44, 62 MB)
Lecture 9, part 2: The notion of Lebesgue integral (1:00:40, 201 MB)
Lecture 10, part 1: Expectation of random variables (46:05, 166 MB)
Lecture 10, part 2: Density of measures and calculating integrals (32:59, 114 MB)
Lecture 11, part 1: Integral substitution theorem, expectation of distributions (43:17, 145 MB)
Lecture 11, part 2: Properties of the integral and expectation, with examples (38:26, 125 MB)
Lecture 12, part 1: Description and construction of random variables (51:37, 147 MB)
Lecture 12, part 2: Generalized inverse of distribution funcitons, remark about the Zikkurat method (31:35, 81 MB)
Lecture 13, part 1: Types of convergence of functions and random variables (42:58, 135 MB)
Lecture 13, part 2: Convergence of functions vs convergence of integrals (36:09, 107 MB)
Lecture 14, part 1: Exchanging limit and integral: Monotone concvergence thm, Dominated convergence thm, Fatou's lemma (53:29, 153 MB)
Lecture 14, part 2: Example of dominated convergence: continuity and differentiability of characteristic functions (31:29, 85 MB)
Lecture 15, part 1: Differentiability of the characteristic function continued (38:39, 130 MB)
Lecture 15, part 1: Exchanging integrals: Fubini's theorem (34:49, 119 MB)
Lecture 16, part 1: Weak convergence of random variables, distributions and distribution functions (41:06, 158 MB)
Lecture 16, part 2: Equivalence of notions of weak convergence; relation to strong convergence (35:00, 128 MB)
Lecture 17, part 1: Weak convergence on metric spaces, Skorokhod representation theorem (33:28, 113 MB)
Lecture 17, part 2: Subsequential limits for distribution functions, vague limits, tightness (52:20, 180 MB)
Lecture 18, part 1: Weak convergence and tightness (27:34, 108 MB)
Lecture 18, part 2: Why we like limit theorems, weak convergence and characteristic functions (38:24, 152 MB)

Material that was pre-recorded (not live):
Suggested literature

• [TIP] Draft lecture notes written by the lecturer: notes/.
• [D] R. Durrett: Probability. Theory and Examples. 4th edition, Cambridge University Press, 2010.
• [R2] Rudin: Real and Complex Analysis