# Math 636 Advanced Probability 1

3

### Prerequisite

Math 314 and Math 341; and Math 431 or Stat 370; or equivalents.

### Description

Foundations of the modern theory of probability with applications. Probability spaces, random variables, independence, conditioning, expectation, generating functions, and Markov chains.

## Desired Learning Outcomes

This should be an advanced course in probability and, therefore, clearly distinguishable from an introductory course like Math 431. Furthermore, it is supposed to be a course in the modern theory of probability, which suggests that it should be based on Kolmogorov's measure-theoretic approach or something equivalent.

### Prerequisites

The official prerequisite is multivariable calculus. Other prior courses that will contribute to student success include:

• an introductory course in probability;
• a course in rigorous mathematical reasoning;
• an introductory course in analysis.

### Minimal learning outcomes

Outlined below are topics that all successful Math 543 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

1. Probability spaces
• Sigma-algebras and Borel sets
• Kolmogorov axioms
• Carathéodory's Extension Theorem
• Lebesgue-Stieltjes measure
2. Random variables
• Measurable maps
• Distributions and distribution functions
3. Independence
• Of events and classes of events
• Of random variables
• Borel-Cantelli Lemmas
4. Expectation
• Of arbitrary nonnegative random variables
• Of integrable real-valued random variables
• Of compositions
• Monotone Convergence Theorem
• Uniform integrability and dominated convergence
5. Conditioning
• Probability conditioned on a non-null set
• Expectation conditioned on a sigma-algebra
• Expectation conditioned on a random variable
6. Probability measures on product spaces
7. Strong Law of Large Numbers
8. Central Limit Theorem
9. Convergence of random variables
• Almost sure
• In probability
• L^p
• weak
10. Discrete-time Martingales

### Textbooks

Possible textbooks for this course include (but are not limited to):

• Achim Klenke, Probability Theory: A Comprehensive Course, Springer, 2008.