Course info

This course covers the foundational aspects of probability theory at the basis of economics and finance. The topics are standard for an introductory graduate probability course.

The course will last 18 hours and span 4 days. It will be divided into lectures of 4 and 4.5 hours. The first part is dedicated to constructing the notion of probability and random variables. Afterwards the course focuses on probability distributions: It discusses their measures of location and dispersion, such as expectation and variance, their common families, and transformation methods. The successive part of the course is dedicated to an advanced and thorough treatment of the topic of conditionality. The course concludes by presenting asymptotic results and convergence theorems for random variables on which econometrics (and statistical inference in general) is based.

Lecture 1: Probability spaces
- Probability axioms
- Sigma algebra
- Probability measure
Lecture 2: Random variables
- Definition of a random variable
- Distribution function
- Radon-Nikodym Theorem
- Density function
Lecture 3: Mathematical expectation
- Definition of mathematical expectation
- Moments
- Variance-covariance matrix
- Moment-generating function
Lecture 4: Common random variables
- Common discrete random variables (uniform, geometric, Bernoulli, binomial, Poisson)
- Common continuous random variables (uniform, normal, exponential, gamma, beta)
Lecture 5: Functions of random variables
- Distribution method
- Density method
- Jacobian matrix
Lecture 6: Conditionality
- Conditional probability
- Independence
- Bayes’ rule
- Conditional distribution, density, and expectation
Lecture 7: Asymptomatic analysis
- Pointwise convergence
- Almost sure convergence
- Convergence in probability
- Convergence in LP
- Monotone convergence theorem
- Dominated convergence theorem
Lecture 8: Laws of large numbers
- Markov inequalities
- Weak law of large numbers (WLLN)
- Strong law of large numbers (SLLN)
- Central limit theorem (CLT)
- Delta method

The course’s material can be accompanied by the following suggested readings:
• Casella, G.H. and Berger, R.L. (2002). Statistical Inference. Duxbury/Thomson Learning.

• Wiley J. and Sons (1986). Probability and Measure. Patrick Billingsley.

• Rudin W. (1987). Real and Complex Analysis. McGraw-Hill.

• Williams D. (1991). Probability with Martingales. Cambridge University Pres.

• Bierens, H. J. (2004). Introduction to the mathematical and statistical foundations of econometrics. Cambridge University Press.