Tools of Modern Probability -- fall semester 2023


Subject code: BMETE95AM33

Classes:

• Wednesday 10:15-12:00, room H601
• Thursday 12:15-14:00, room H601

Lecturer: Imre Péter Tóth

QUESTIONS AND ANSWERS session before the fourth exam: Monday, 22 January 2024 at 18:00, ONLINE ONLY (sorry for that, Monday evenings are hard), using Microsoft Teams. To join, use this link. If there is some problem (usually there is), call me at +36205372256


HOMEWORK RESULTS. The corrected homeworks can be found in the Moodle system of the Faculty of Natural Sciences (for those to whom I could not give them in person). Each exercise is evaluated with a CODE 0,1,2 or 3 (which is not a score itself). Meaning of the code:

• 3 means "correct",
• 2 means "solved with some mistake",
• 1 means "started in a correct direction but not solved",
• 0 means "wrong".

These codes are converted to homework scores using the following rule:

• "correct" exercises are worth 5 scores,
• "solved with some mistake" exercises are worth 4 scores,
• the rest is worth 0 scores,
• only the best 2/3 (which is 8 exercises) are taken into account.

So the maximum score is 40 and 20 are needed to obtain the signature for the course.

VOLUNTARY HOMEWORK for extra score, from the updated practice exercise sheet:
Exercise 7.2, 8.2, 8.4/c,d,e,i,l, 9.1, 10.11
Submission: You can bring your homeworks to an exam, of submit them online in pdf using the Moodle system of the Faculty of Natural Sciences.
Deadline: bring it to the exam at the latest. WARNING: if you only bring your HW to the exam, I may not be able to grade it immediately, so you may not get your grade on the spot.

EXAMS:

• 20 December 2023 (Wednesday) 9:00, room H607
• 10 January 2024 (Wednesday) 9:00, room H607
• 17 January 2024 (Wednesday) 9:00, room H607
• 24 January 2024 (Wednesday) 9:00, room H607

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.
Note: from Section 10 of the exercise sheet very little was covered, so most exercises there are for your fun only: don't worry about them.

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]

HOMEWORKS to hand in from the updated practice exercise sheet:
Exercise 1.2, 1.5, 2.3, 2.5, 3.6, 5.2, 5.4, 5.7, 6.2/b,c, 6.6/f, 6.7, 6.8
Submission: You can bring your homeworks to an exam, of submit them online in pdf using the Moodle system of the Faculty of Natural Sciences.
Deadline: bring it to the exam at the latest. WARNING: if you only bring your HW to the exam, I may not be able to grade it immediately, so you may not get your grade on the spot.

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 some 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.

Grading rules:

• Over the course, the maximum total score is 100: 40 for the homeworks and 60 for the final oral exam.
• Only the best 2/3 of the homeworks 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.)

Preliminary plan of topics (most of which were not 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):

• Linear spaces, linear transformations (definition, examples). See
  • lecture notes 10: Linear spaces and transformations
  • videos
    • 20: Linear spaces (27:38, 85 MB)
    • 21: Linear transformations, bilinear forms (20:13, 62 MB)
• Inner product spaces, normed spaces (definition, examples). See
  • lecture notes 11: Inner product spaces, normed spaces
  • videos
    • 22: Normed spaces (34:36, 106 MB)
    • 23: Inner product spaces (22:57, 70 MB)
    • 24: L^2 is an inner product space (11:47, 37 MB)
    • 25: Inner product and norm (21:19, 67 MB)
• Banach spaces, Hilbert spaces (definitions, example). See
  • lecture notes 12: Banach spaces, Hilbert spaces
  • videos
    • 26: Banach space, Hilbert space: definition (14:52, 52 MB)
    • 27: Completeness of L^2 and L^p: idea of proof (16:39, 61 MB)
    • 28: Proof of completeness of L^2 (25:25, 94 MB)
    • 29: Hilbert spaces and separability (11:15, 40 MB)
• Riesz representation theorem. See
  • lecture notes 13: Riesz representation theorem
  • videos
    • 30: Riesz representation theorem: statement (37:13, 116 MB)
    • 31: A theorem about closed convex sets (31:49, 94 MB)
    • 32: Proof of the Riesz representation theorem (24:41, 74 MB)
• Radon-Nikodym theorem. See
  • lecture notes 14: Radon-Nikodym theorem
  • videos
    • 33: Radon-Nikodym theorem, statement (20:21, 67 MB)
    • 34: Radon-Nikodym theorem, idea of the proof (13:45, 48 MB)
    • 35: Radon-Nikodym theorem, details of the proof (30:16, 105 MB)
• Conditional expectation. See
  • lecture notes Conditional expectation: introduction, definition, existence, some basic properties
  • videos
    • 36: Conditional expectation: introduction 1 - why it's hard (26:16, 92 MB)
    • 37: Conditional expectation: introduction 2 - motivation (12:00, 36 MB)
    • 38: Conditional expectation: idea of the construction (33:26, 103 MB)
    • 39: Conditional expectation: definition (25:17, 77 MB)
    • 40: Conditional expectation: first example (07:48, 23 MB)
    • 41: Conditional expectation: uniqueness (04:33, 13 MB)
    • 42: Conditional expectation: existence (12:58, 39 MB)

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