Summary

Introductory graduate course on probability and measure theory with applications to economics and finance. The course presents probability concepts with the aim of enhancing familiarity with mathematical thinking and formal proof techniques.

Lecture Notes

[links without solutions] [link with solutions]

Slides

L1: Probability space and random variables [slides]
L2: Expectation and common random variables [slides]
L3: Functions of random variables and conditionality [slides]
L4: Limiting distributions, laws of large numbers, and Central Limit Theorem [slides]

Handouts

Stochastic Calculus
Linear Regression – MLE and Asymptotics

Exercises

Exercise sheet 1
Exercise sheet 2 – [solutions and hints]
Exercise sheet 3

Suggest a Topic

References

Main reference:
Durrett, R. (2019). Probability: Theory and Examples (5th ed.). Cambridge University Press.

Other references:
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.

Syllabus

Probability spaces:
Probability axioms. Sigma-algebra. Probability measure.

Random variables:
Definition of a random variable. Distribution function. Radon-Nikodym Theorem. Density function

Mathematical expectation:
Definition of mathematical expectation. Moments. Variance-covariance matrix. Moment-generating function

Common random variables:
Common discrete random variables (uniform, geometric, Bernoulli, binomial, Poisson). Common continuous random variables (uniform, normal, exponential, gamma, beta)

Functions of random variables:
Distribution method. Density method. Jacobian matrix

Conditionality:
Conditional probability. Independence. Bayes’ rule. Conditional distribution, density, and expectation

Asymptotic analysis:
Pointwise convergence. Almost sure convergence. Convergence in probability. Convergence in 𝐿𝑝. Monotone convergence theorem. Dominated convergence theorem.

Laws of large numbers:
Markov inequalities. Weak law of large numbers (WLLN). Strong law of large numbers (SLLN). Central limit theorem (CLT). Delta method