The lecture notes for this course are courtesy of Brenda Ng, a student in the class. Used with permission.
Lecture notes files.
| LEC # |
TOPICS |
| 1 |
Probability Spaces, Properties of Probability (PDF) |
| 2 |
Random Variables (PDF) |
| 3 |
Expectation, Kolmogorov's Theorem about Consistent Families of Distributions (PDF) |
| 4 |
Kolmogorov's Theorem (cont.), Law of Large Numbers (PDF) |
| 5 |
Strong Law of Large Numbers (PDF) |
| 6 |
Bernstein's Polynomials, Problem of Moments, de Finetti's Theorem (PDF) |
| 7 |
0-1 Laws (PDF) |
| 8 |
Convergence of Random Series, Stopping Time, Wald's Identity, Another Proof of SLLN (PDF) |
| 9 |
Convergence of Laws (PDF) |
| 10 |
Convergence of Laws (cont.), Tightness (PDF) |
| 11 |
Characteristic Functions, Central Limit Theorem (PDF) |
| 12 |
Multivariate Gaussian Laws and CLT (PDF) |
| 13 |
Lindeberg's CLT, Levy's Equivalence Theorem, Three Series Theorem (PDF) |
| 14 |
Levy's Continuity Theorem, Poisson Approximation, Conditional Expectations (PDF) |
| 15 |
Properties of Conditional Expectation, Martingales, Doob's Decomposition (PDF) |
| 16 |
Uniform Integrability, Right-closable Martingales, Optional Stopping (PDF) |
| 17 |
Inequalities, Convergence of (Sub)martingales (PDF) |
| 18 |
Convergence of (Sub)martingales (cont.), Fundamental Wald's Identity (PDF) |
| 19 |
Portmanteau Theorem, Lipschitz Functions (PDF) |
| 20 |
Stone-Weierstrass Theorem, Levy-Prohorov Metric (PDF) |
| 21 |
Metrics for Convergence of Laws (PDF) |
| 22 |
Tightness and Uniform Tightness (PDF) |
| 23 |
Strassen's Theorem (PDF) |
| 24 |
Strassen's Theorem (cont.) (PDF) |
| 25 |
Relationship between Ky Fan Metric and Prohorov Metric, Kantorovich-Rubinstein Theorem (PDF) |
| 26 |
Prekopa-Leindler Inequality (PDF) |
| 27 |
Prekopa-Leindler Inequality (cont.), Transportation Cost Inequality, Concentration of Gaussian Measure (PDF) |
| 28 |
Stochastic Processes, Kolmogorov's Theorem, Brownian Motion (PDF) |
| 29 |
Brownian Motion (cont.), Brownian Bridge, Application of Brownian Bridge, Kolmogorov-Smirnov Test (PDF) |
| 30 |
Donsker Invariance Principle (PDF) |
| 31 |
Kolmogorov Chaining (PDF) |
| 32 |
Kolmogorov Chaining (cont.), Strong Markov Property of Brownian Motion (PDF) |
| 33 |
Reflection Principles (PDF) |
| 34 |
Laws of Brownian Motion at Markov Times, Skorohod's Imbedding (PDF) |
| 35 |
Law of Iterated Logarithm (PDF) |