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# Coursebooks

## Advanced probability and applications

#### COM-417

#### Lecturer(s) :

Lévêque Olivier#### Language:

English

#### Summary

In this course, various aspects of probability theory are considered. The first part is devoted to the main theorems in the field (law of large numbers, central limit theorems), while the second part focuses on the theory of martingales in discrete time.#### Content

I. Probability

- sigma-fields, probability measures, random variables

- independence, expectation

- convergence of sequences of random variables

- laws of large numbers- central limit theorem

- concentration inequalities

- moments

II. Martingales

- conditional expectation

- definition and properties of a martingale

- stopping times, optional stopping theorem

- maximal inequalities

- convergence theorems

#### Keywords

probability, measure theory, martingales, convergence theorems

#### Learning Prerequisites

##### Required courses

Basic probability course

Calculus courses

##### Recommended courses

complex analysis

##### Important concepts to start the course

This course is NOT an introductory course on probability: the students should have a good understanding and practice of basic probability concepts such as: distribution, expectation, variance, independence, conditional probability.

The students should also be at ease with calculus. Complex analyisis is a plus, but is not required.

On the other hand, no prior background on measure theory is needed for this course: we will go through the basic concepts one by one at the beginning.

#### Learning Outcomes

By the end of the course, the student must be able to:- Understand the foundations of probability theory
- Acquire a solid knowledge of martingale theory

#### Teaching methods

Ex cathedra + exercises

#### Expected student activities

active participation to exercise sessions

#### Assessment methods

Midterm 10%, homeworks 10%, exam 80%

#### Resources

##### Bibliography

Sheldon M. Ross, Erol A. Pekoz, A Second Course in Probability,1st edition, www.ProbabilityBookstore.com, 2007.

Jeffrey S. Rosenthal, A First Look at Rigorous Probability Theory,2nd edition, World Scientific, 2006.

Geoffrey R. Grimmett, David R. Stirzaker, Probability and Random Processes,3rd edition, Oxford University Press, 2001.

Richard Durrett, Probability: Theory and Examples,4th edition, Cambridge University Press, 2010.

##### Ressources en bibliothèque

- A Second Course in Probability / Ross
- A First Look at Rigorous Probability Theory / Rosenthal
- Probability and Random Processes / Grimmett
- Probability: Theory and Examples / Durrett

##### Notes/Handbook

available on the course website

##### Websites

#### Prerequisite for

Advanced classes requiring a good knowledge of probability

### In the programs

**Semester**Spring**Exam form**Written**Credits**

6**Subject examined**

Advanced probability and applications**Lecture**

3 Hour(s) per week x 14 weeks**Exercises**

2 Hour(s) per week x 14 weeks

**Semester**Spring**Exam form**Written**Credits**

6**Subject examined**

Advanced probability and applications**Lecture**

3 Hour(s) per week x 14 weeks**Exercises**

2 Hour(s) per week x 14 weeks

### Reference week

Mo | Tu | We | Th | Fr | |
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8-9 | |||||

9-10 | |||||

10-11 | |||||

11-12 | |||||

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16-17 | |||||

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19-20 | |||||

20-21 | |||||

21-22 |

### legend

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