# NMSA405 - Probability theory 2

**Thursday 12.20 - 13.50 K2**

#### SIS

#### Course Notes - version 4. 1.

## Requirements for obtaining the course credit:

- at least 1 homework presented at the blackboard
- active participation (at most 3 absences)

## 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 8 topics and then has some time to prepare the answer (it could be in English,
in Czech or combination of both).

Afterwards some additional questions (from any topic covered during the semester) could be asked.

### Topics

1. random sequence, product σ-algebra, random element with values in the space of real sequences,
distribution uniquelly determined by finite-dimensional distributions, Daniell's extension theorem

2. stopping time, filtration, stopping time σ-algebra, examples, properties and relations,
strong Markov property of a random walk

3. (sub)martingale, definition and examples, stability of (sub)martingale property with respect to filtration
and convex transformation, Doob decomposition theorem

4. optional stopping theorem, optional sampling theorem, Wald's equations (general version)

5. supermartingale goes bankrupt forever, Doob's maximal inequalities, Kolmogorov inequality

6. Doob's upcrossing inequality, submartingale convergence theorems (a.s., in L_{1})

7. backwards (sub)martingale, convergence thereoms (a.s., in L_{1}),
submartingale converges or explodes

8. limit theorems for martingale differences

### 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

5.1., 19.1., 29.1., 7.2., 15.2.