Lecture Notes

All Lecture Notes in One File (PDF - 1.9 MB)

Note: A more recent version of this course, taught by Prof. Dmitry Panchenko at Texas A&M University, is available here. Updated Lecture Notes include some new material and many more exercises.

LEC # TOPICS NOTES
1 Probability spaces, properties of probability (PDF)
2-3 Random variables and their properties, expectation (PDF)
4 Kolmogorov's theorem about consistent distributions (PDF)
5 Laws of large numbers (PDF)
6 Bernstein's polynomials, Hausdorff and de Finetti theorems (PDF)
7 0-1 laws, convergence of random series (PDF)
8

Stopping times, Wald's identity

Markov property, another proof of SLLN

(PDF)
9-10 Convergence of laws, selection theorem (PDF)
11 Characteristic functions, central limit theorem on the real line (PDF)
12 Multivariate normal distributions and central limit theorem (PDF)
13

Lindeberg's central limit theorem

Levy's equivalence theorem, three series theorem

(PDF)
14

Levy's continuity theorem

Levy's equivalence theorem, three series theorem (cont.)

Conditional expectation

(PDF)
15-16

Martingales, Doob's decomposition

Uniform integrability

(PDF)
17 Optional stopping, inequalities for Martingales (PDF)
18-19 Convergence of Martingales (PDF)
20-21

Convergence on metric spaces, Portmanteau theorem

Lipschitz functions

(PDF)
22 Metrics for convergence of laws, empirical measures (PDF)
23 Convergence and uniform tightness (PDF)
24-25 Strassen's theorem, relationship between metrics (PDF)
26-27 Kantorovich-Rubinstein theorem (PDF)
28-29 Prekopa-Leindler inequality, entropy and concentration (PDF)
30 Stochastic processes, Brownian motion (PDF)
31 Donsker invariance principle (PDF)
32-33 Empirical process and Kolmogorov's chaining (PDF)
34-35 Markov property of Brownian motion, reflection principles (PDF)
36

Laws of Brownian motion at stopping times

Skorohod's imbedding

(PDF)
37 Laws of the iterated logarithm (PDF)