# Advanced probability and applications

## 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 theorem, concentration inequalities), while the second part focuses on the theory of martingales in discrete time.

## Content

- sigma-fields, random variables

- probability measures, distributions

- independence, convolution

- expectation, characteristic function

- random vectors and Gaussian random vectors

- inequalities, convergences of sequences of random variables

- laws of large numbers, applications and extensions

- convergence in distribution, central limit theorem and applications

- moments and Carleman's theorem

- concentration inequalities

- conditional expectation

- martingales, stopping times

- martingale convergence theorems

## Keywords

probability theory, 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 main ideas at the heart of probability theory

## Teaching methods

Ex cathedra lectures + exercise sessions

## Expected student activities

active participation to exercise sessions

## Assessment methods

- graded homeworks : 25%

- Written exam on campus during the exam period : 75%

## 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.

Patrick Billingsley, Probability and Measure, 3rd edition, Wiley, 1995.

## Ressources en bibliothèque

- Sheldon M. Ross, Erol A. Pekoz, A Second Course in Probability, 1st ed
- Jeffrey S. Rosenthal, A First Look at Rigorous Probability Theory, 2nd ed
- Richard Durrett, Probability: Theory and Examples, 4th ed
- Patrick Billingsley, Probability and Measure, 3rd ed

## 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 (summer session)**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 (summer session)**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 (summer session)**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 (summer session)**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 (summer session)**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 (summer session)**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 (summer session)**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 (summer session)**Subject examined:**Advanced probability and applications**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Exam form:**Written (summer session)**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 (summer session)**Subject examined:**Advanced probability and applications**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Exam form:**Written (summer session)**Subject examined:**Advanced probability and applications**Lecture:**3 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Exam form:**Written (summer session)**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

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