Lecture Notes on Probability

Disclaimer 3 1. Why we need measure theory 4 1.1. Riddle 1 4 1.2. Riddle 2 4 1.3. Riddle 3 4 1.4. Why we need measure theory 4 2. Measure theory 6 2.1. π-systems, algebras and sigma-algebras 6 3. Carathéodory’s Theorem and constructing measures 8 4. Events and random variables 10 5. Independence 13 6. The tail sigma-algebra 16 7. Expectations 19 8. A strong law of large numbers and the Chernoff bound 22 9. The weak law of large numbers 25 10. Conditional expectations 27 10.1. Why things are not as simple as they seem 27 10.2. Conditional expectations in finite spaces 27 10.3. Conditional expectations in L 27 10.4. Conditional expectations in L 28 10.5. Some properties of conditional expectation 29 11. The Galton-Watson process 30 12. Markov chains 33 13. Martingales 37 14. Stopping times 41 15. Harmonic and superharmonic functions 43 16. The Choquet-Deny Theorem 46 17. Characteristic functions and the Central Limit Theorem 48