NMSA405 - Probability theory 2

Thursday   14.00 - 15.30   K5

SIS


Video recordings of last year's lectures and some other materials are available in the platform Moodle after enrolment.

Course Notes - version 5. 10. 2021


Requirements for obtaining the course credit:

Instructions for the final exam:

The exam is oral. All material covered during the lecture could be part of the exam.
First the student draws one of the following 20 topics and then has some time to prepare the answer (it could be in English, in Czech or combination of both). Each topic consists of the formulation of theorem and presentation of its proof (part a) and related definitions and basic properties (part b).
Afterwards some additional questions (from any topic covered during the semester) could be asked. A minor derivation or calculation may be required to examine the understanding of the substance. It does not necessarily have to be a derivation that was part of the lecture, it could have appeared in exercise classes or it can be obtained by a simple independent reasoning.

Topics

1a. The probability distribution of a random sequence is uniquely determined by finite dimensional distributions.
1b. Random sequence, product σ-algebra, random element with values in the space of real sequences, finite dimensional set.

2a. Daniell's extension theorem.
2b. Random sequence, projective family of probability distributions, product σ-algebra, product measure.

3a. Strong Markov property of a random walk. Stationarity with respect to a stopping time.
3b. Random walk, stopping time and its properties, filtration, stopping time σ-algebra.

4a. Stability of the martingale property with respect to filtration. Convex transformation of a (sub)martingale.
4b. Martingale, submartingale, filtration, example of a martingale.

5a. Doob decomposition theorem.
5b. Martingale, submartingale, predictable sequence, compensator, example of a submartingale.

6a. Optional stopping theorem.
6b. Stopping time and its properties, stopping time σ-algebra, martingale, submartingale.

7a. Optional sampling theorem for two bounded stopping times.
7b. Stopping time and its properties, stopping time σ-algebra, martingale, submartingale.

8a. Optional sampling theorem.
8b. Stopping time and its properties, stopping time σ-algebra, martingale, submartingale.

9a. Wald's equations - general and basic version.
9b. Random walk, stopping time and its properties, first exit time.

10a. Supermartingale goes bankrupt forever.
10b. Martingale, supermartingale, stopping time σ-algebra, first exit time, example of a supermartingale.

11a. Doob's maximal inequalities. Kolmogorov's inequality.
11b. Martingale, submartingale, examples of martingales and submartingales.

12a. Doob's upcrossing inequality.
12b. Martingale, submartingale, stopping time and its properties, example of a submartingale.

13a. Doob's submartingale convergence theorem.
13b. Martingale, submartingale, examples of martingales and submartingales.

14a. Doob's backwards submartingale convergence theorem.
14b. Backwards martingale, backwards submartingale, example of a backwards martingale.

15a. Convergence of uniformly integrable (sub)martingale.
15b. Martingale, submartingale, uniform integrability, example of a uniformly integrable martingale.

16a. Optional sampling theorem for uniformly integrable martingale.
16b. Martingale, stopping time and its properties, stopping time σ-algebra, example of a uniformly integrable martingale.

17a. Continuity of the conditional expectation with respect to the condition.
17b. Conditional expectation, filtration, martingale, backwards martingale, uniform integrability.

18a. Submartingale converges or explodes.
18b. Martingale, submartingale, first exit time, example of a submartingale.

19a. Summability of martingale differences. Strong law of large numbers for martingale differences.
19b. Martingale, martingale difference sequence, example of a martingale difference sequence.

20a. Central limit theorem for martingale differences.
20b. Martingale, martingale difference sequence, example of a martingale difference sequence.

Grading

1 (excellent): student knows the proofs, clearly understands the material and is able to use it

2 (very good): slight flaws

3 (good): student knows only simple proofs or has problems with the explanation of theoretical results and their application

4 (failed): student is unable to correctly formulate some definition or theorem (even if other answers are satisfactory) or shows lack of understanding of the material

Dates

7.1., 11.1., 19.1., 26.1., 27.1., 1.2., 3.2., or possibly by mutual agreement