MATH-F-302

• Contenu du cours

This is an advanced (measure theory based) course on probability theory.
This is the tentative table of contents:

1. Recap

1.1. Probability spaces.
1.2. Random variables.
1.3. Distribution functions.
1.4. Moments.
1.5. Random vectors and sequences.
1.6. Modes of convergence.

2. Infinite sequences of independent random variables

2.1. IID sequences.
2.2. Independent systems of events.
2.3. Independent random variables.
2.4. Construction of an independent random sequence.

3. Almost sure statements

3.1. Almost sure convergence.
3.2. Borel-Cantelli Lemmas.
3.3. Kolmogorovís 0-1 law.
3.4. Strong law of large numbers.

4. Conditional expectations

4.1. Conditioning on a discrete random variable.
4.2. The general conditional expectation.
4.3. L2 version of the conditional expected value.

5. Martingales

5.1. Martingales.
5.2. Basic properties of a martingale.
5.3. Betting strategies.
5.4. Stopping times.
5.5. A martingale CLT.

6. Brownian motion

6.1. History.
6.2. A discrete model.
6.3. Brownian motion.
6.4. Existence.
6.5. Donskerís invariance principle.
6.6. Properties.
6.7. Hitting times.
6.8. Reflection principle.

Literature

Billingsley (1995). Probability and Measure, Wiley.

• Exercises.

Sheet 1
Sheet 2
Sheet 3
Sheet 4
Sheet 5
Sheet 6
Sheet 7


Updated: Apr 20, 2017