Tools of Modern Probability -- fall semester 2021

Subject code: BMETE95AM33

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

HOMEWORK RESULTS: The results of the homeworks and the calculated homework score can be found here. The list is anonymous, the names are substituted by a unique identifier called "ID". Each students can find her/his "ID" in Moodle. (Sorry for this inconvenience. The gradebook of Moodle is not well suited for such calculations.) I hope that all homework results are administered. If something is missing or you find errors, please let me know!

Calculating the homework score
• The solution to each homework is evaluated with a "code" with the following meaning:
• "3" means correctly solved
• "2" means solved with some error
• "1" means started on a correct track, but not solved
• "0" means completely wrong.
At the end of the semester these codes are translated to a homework score. At the translation, the big difference is between "solved" and "not solved", so a "correctly solved" is worth 1 point, a "solved with some error" is worth 0.8 points, and the rest is worth no points at all.
• Only the best 10 homeworks are taken into account.
• Eventually, the score is scaled to be between 0 and 40 (a multiplication by 4).
EXAMS:
• 22 December 2021 (Wednesday) 10:00, room ???
• 06 January 2022 (Thursday) 10:00, room ???
• 13 January 2022 (Thursday) 10:00, room ???
• 20 January 2022 (Thursday) 10:00, room ???
The exam will be ORAL. Please register for the exam in Neptun. When the registration closes (at noon before the day of the exam), I will decide on the order of the registered students, and let you know when it will be your turn. This may be up to 4 hours after the start of the exam. If your time is limited, please let me know, so that I can take it into account. Also: I'm flexible about exam times. If the exams announced in Neptun are not good for you, contact me, and we try to find a suitable time.

On the exam, every student gets at least one calculation exercise (similar to those in the exercise sheets), and two randomly chosen theoretical questions from the following list. NOTE: Before the exam, everybody can choose two of these questions, which she/he would not like, to exclude from the list.
1. Gaussian integrals; spherically symmetric integrals, surface of hyperspheres [notes/01, section 1-2]
2. Almost Gaussian integrals, Laplace's method [notes/01, section 3]
3. Euler gamma function, Stirling's approximation [notes/01, section 4-5]
4. Measure space, probability space. Push-forward of measures. Distribution of random variables [notes/02, section 1.1, 1.2; D, section 1.1-1.5]
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. [notes/02, section 1.3.1; D, section 1.6]
6. Construction of random variables with a given distribution, generalized inverse of distribution functions [D, Theorem 1.2.2]
7. Exchanging the integral and the limit: monotone convergence theorem, dominated convergence theorem, Fatou lemma. [notes/02, section 1.3.2; D, section 1.6.2]
8. Characteristic functions of random variables, characteristic functions of probability distributions. Continuity, differentiability, continuous differentiability of the characteristic function. [Exercises 2.10,...,2.14]
9. Product space, product measure. Exchanging integrals: Fubini's theorem [D, section 1.7]
10. Weak convergence of random variables, weak convergence of probability distributions. Equivalent descriptions, relation to strong convergence. [D, Theorem 3.2.2, Theorem 3.2.3]
11. Subsequential limits for distribution functions, vague limits, tightness. [D, Theorem 3.2.6, Theorem 3.2.7]
12. Hilbert spaces - completeness of L^2 [notes/10,11,12]
13. Riesz representation theorem [notes/13]
14. Absoulte continuity. Radon-Nikodym theorem. [notes/14]
15. Conditional expectation of random variables. Definition, existence, uniqueness [notes/15]

Paractice exercises Here.

HOMEWORK 1 due on 13 October 2021: Here.

HOMEWORK 2 due on 17 November 2021: exercises 2.3, 5.4 and 5.7 from the practice exercise sheet.

HOMEWORK 3 due on 24 November 2021: exercises 6.6/e,f; 6.7 and 6.8 from the practice exercise sheet. In parts e.) and f.) of exercise 6.6 it is OK to use the formula for the derivative obtained in class (or in parts c.) and d.) ). The task now is only to show continuity of that derivative.

HOMEWORK 4 due on 02 December 2021: exercises 7.1/d,e; 7.2, 7.3/e,f; 8.4/g,i,j from the practice exercise sheet.

HOMEWORK 5 due on the exam: exercises 10.12, 10.13, 10.14 from the practice exercise sheet.

ERROR REPORT REQUEST: The lecture notes and videos certainly contain errors. I am grateful to anyone finding and reporting these errors. Please let me know of every error you find!

Lecturer: Imre Péter Tóth

Course organization:
• There will be lectures, homeworks and an oral final exam. The homeworks contribute to the grade with 40% weight.
• Lectures will be held at the university as long as this is possible. I suggest attending the lectures for everybody who can, but there is no obligation.
• Detailed lecture notes and lecture videos will be available online, on the course home page.
• Homeworks will be published electronically on the coure home page, and must be submitted electronically.
• Results of the homeworks will be published on the course home page (in anonym form), and corrected homeworks will be sent back to the students electronically.
• There will be regular question sessions online (via video conference using MS Teams) for everybody.
• The lectures consist of theoretical and practice parts, but these will not be strictly separated in time.
• Students can hopefully take the final in person.
• 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 last year 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