# MA8704 Probability Theory and Asymptotic Methods (Fall 2018)

## Messages

- [21/11] The Wednesday exam will also be in room 922 SB2.
- [20/11] Note that room for the examination on November 28 is at this time not settled.
- [20/11] As discussed in the lecture today, the exam will be Tuesday and Wednesday November 27 and 28. Further information will be given on the course webpage.
- [20/10] A note with a more detailed proof of the Monotone Class Theorem (Theorem 6.2 in book) can now be downloaded from "Lectures", week 37, on this webpage.
- [19/10] In the lectures yesterday we decided to have the (oral) exam on Tuesday November 27.
- [27/08] I will be out of town on Tuesday September 4 an will therefore have to cancel the lecture on this day.
- [23/08] In the meeting on August 21 we agreed on the following lecture schedule: Tuesdays 08:15-10:00 in Room 656, Sentralbygg 2 (Simastuen), and Thursdays 16:15-18:00 in Room 656, Sentralbygg 2 (Simastuen).
- [12/06] Welcome to the home page of MA 8704 Probability Theory and Asymptotic Methods. Our first meeting will be Tuesday August 21, 14:15-15:00 in Room 656, Sentralbygg II. The main point of this meeting is to agree on a schedule for lectures. A brief overview of the course will also be given. If you are not able to attend this meeting, but plan to take the course, please send an email to bo [dot] lindqvist [at] ntnu [dot] no.

## Textbooks

Jean Jacod and Philip Protter. Probability Essentials. Second Edition. Springer-Verlag Berlin Heidelberg, 2004.

The book can be downloaded from NTNU freely from Springer Link:

https://link.springer.com/content/pdf/10.1007%2F978-3-642-55682-1.pdf

*Preface to the First Edition of the book:*

We present here a one semester course on Probability Theory. We also treat measure theory and Lebesgue integration, concentrating on those aspects which are especially germane to the study of Probability Theory. The book is intended to fill a current need: there are mathematically sophisticated students and researchers (especially in Engineering, Economics, and Statistics) who need a proper grounding in Probability in order to pursue their primary interests. Many Probability texts available today are celebrations of Probability Theory, containing treatments of fascinating topics to be sure, but nevertheless they make it difficult to construct a lean one semester course that covers (what we believe) are the essential topics.

Chapters 1–23 provide such a course. We have indulged ourselves a bit by including Chapters 24–28 which are highly optional, but which may prove useful to Economists and Electrical Engineers.

*Book used in previous versions of the course:*

Alan F. Karr. Probability, 1993, Springer Verlag.

## Curriculum

The curriculum is basically what has been gone through in lectures and exercises (see "Lectures" below). Some details and comments to each chapter are given below.

From Jacod and Protter:

- Chapter 1: Introduction, known from elementary courses.
- Chapter 2: Axioms, basic definitions (important).
- Chapter 3-5: Mostly known from elementary courses.
- Chapter 6: Except Theorem 6.3, only result of Theorem 6.4. Proof of Monotone Class Theorem in separate note (week 37).
- Chapter 7: Only first paragraph of proof of Theorem 7.2 ("only if").
- Chapter 8: Measurability is an important key word here.
- Chapter 9: Important chapter, you should have some knowledge of the proofs.
- Chapter 10: You should know the results, but no details of proofs.
- Chapter 11-12: Some of this is known from earlier courses, e.g. Jacobi transformations.
- Chapter 13-14: Skip proof of Theorem 14.1. Have a look at Ch. 6 of Karr.
- Chapter 15: You should have some knowledge on the contents.
- Chapter 16: Gaussian distribution: Much of this is known from other courses.
- Chapter 17: Proof of Theorem 17.5 can be skipped. Important chapter.
- Chapter 18-19: Skip the proof of Theorem 18.7. Important chapter.

From Karr, Chapter 7:

- Section 7.1: Lemma 7.1, Theorem 7.2 first part, and Theorem 7.3; Sections 7.2 and 7.3; Sections 7.5.1-7.5.3.
- Knowledge of main ideas of exercises studied in the lectures.

## Corrections to textbooks

## Lectures

Week | Topic | Textbook | Exercises | Comment | ||
---|---|---|---|---|---|---|

35 | Axioms of probability | Ch. 1-2 | 2.1, 2.2, 2.3, 2.6, 2.7, 2.13, 2.15, 2.17 | Tuesday: Ch. 2 (Thm.2.4 remains). Thursday: Finished Ch. 2 | ||

36 | Conditional probabilities, probabilities and random variables on countable spaces. | Ch. 3-6 | 3.1, 3.6, 3.7, 3.9, 3.17, 3.18, 5.20, 5.21 | No lecture on Tuesday 4 Sept. Read yourself Ch. 3-5. Thursday: Discussion of Ch.1-5 plus exercises. Preview of Ch. 6. | ||

37 | Construction of probability measures. Random variables. | Ch. 6-8 | 7.1, 7.2*,7.11, 7.12, 7.13, 7.14, 7.15, 7.17, 7.18 | Tuesday: Finished discussion of Ch 6-7. Thursday: Ch. 8 plus exercises. Proof of Monotone Class Theorem. | ||

38 | Integration with respect to a probability measure. | Ch. 9 | 9.1, 9.5, 9.6, 9.7, 9.8, 9.9 | Tuesday: Finished Ch. 8, Started Ch. 9. Thursday: Discussed the rest of Ch. 9, in particular the result of Theorem 9.1, plus exercises. | ||

39 | Independent random variables. Probability distributions on R and Rn | Ch. 10-12 | 10.3, 10.5, 10.11, 10.12, 10.17, 10.18 | Note on zero-one-laws | ||

40 | Characteristic functions | Ch. 13-14 | Extra 1.1, 11.13, 11.14, 11.15, 12.7, 12.12, 12.14 | Ch. 6 in Karr. See also Wikipedia | ||

41 | Sums of independent random variables. Gaussian random variables. | Ch. 15-16 | 14.4, 14.5, 14.6, 14.7, 14.8, 14.11, 14.18 | Tuesday: Went through Ch. 15-16. On Thursday we will discuss exercises only. A few remain from last week, e.g. from Ch. 12. | ||

42 | Review of chapters 1-16. Convergence of random variables. | Ch. 1-16 + Ch. 17 | 15.3, 15.4, 15.5, 15.6, 15.11, 15.12*, 16.1, 16.2, 16.3, 16.6. | Tuesday: Review lecture (Ch. 1-16). Thursday: Started Ch. 17 (to middle of p. 144). | ||

43 | Weak convergence | Ch. 17-18 | Extra 2.1, 2.2, 17.1, 17.3, 17.4, 17.6, 17.9, 17.13, 17.14 | Tuesday: Finished discussion of Ch. 17. Started Ch. 18, until Theorem 18.4. Thursday: Discussed until Theorem 18.6. | ||

44 | Weak convergence and characteristic functions. | Ch. 19 | 18.1, 18.4, 18.5, 18.6, 18.8, 18.14 | Tuesday: Finished Ch. 18. Thursday: Ch. 19 + exercises. | ||

45 | Classical limit theorems | Ch. 7 in Karr | 19.1, 19.2, 19.3. | Chapter 7 in Karr's book covers several applications of probability theory that are relevant for statisticians, e.g., strong law of large numbers, central limit theorems and asymptotic theory for the maximum likelihood estimator. Tuesday: Section 7.1: Lemma 7.1, Theorem 7.2 first part, and Theorem 7.3; all of Section 7.2 . Thursday: Section 7.3. (Note corrections to Karr's book). | ||

46 | Classical limit theorems (continued) | Ch. 7 in Karr | Extra 3.1, 3.2, Karr: 7.2, 7.3, 7.9, 7.12 | Karr: 7.5.1-7.5.3. | ||

47 | Last week of lectures! |

## Lecturer

Professor Bo Lindqvist. Office: 1129, Sentralbygg 2. Email: bo [dot] lindqvist [at] ntnu [dot] no

## Schedule

- Tuesdays 08:15-10:00, Room 656, SB2 (Simastuen). First time: 28 August.
- Thursdays 16:15-18:00, Room 656, SB2 (Simastuen). First time: 30 August.

## Exam

Oral exam will be held on Tuesday November 27 and Wednesday November 28.