This is an archived course. A more recent version may be available at ocw.mit.edu.

Calendar

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

Stopping times, Wald's identity

Markov property, another proof of SLLN

 Problem set 1 out
9-10 Convergence of laws, selection theorem Problem set 1 due in Lec #9
11 Characteristic functions, central limit theorem on the real line  
12 Multivariate normal distributions and central limit theorem  
13

Lindeberg's central limit theorem

Levy's equivalence theorem, three series theorem

  
14

Levy's continuity theorem

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

Conditional expectation

 Problem set 2 out
15-16

Martingales, Doob's decomposition

Uniform integrability

 Problem set 2 due in Lec #15
17 Optional stopping, inequalities for Martingales  
18-19 Convergence of Martingales Problem set 3 out in Lec #19
20-21

Convergence on metric spaces, Portmanteau theorem

Lipschitz functions

 Problem set 3 due in Lec #20
22 Metrics for convergence of laws, empirical measures  
23 Convergence and uniform tightness  
24-25 Strassen's theorem, relationship between metrics  
26-27 Kantorovich-Rubinstein theorem  
28-29 Prekopa-Leindler inequality, entropy and concentration Problem set 4 out in Lec #29
30 Stochastic processes, Brownian motion Problem set 4 due
31 Donsker invariance principle  
32-33 Empirical process and Kolmogorov's chaining  
34-35 Markov property of Brownian motion, reflection principles  
36

Laws of Brownian motion at stopping times

Skorohod's imbedding

  
37 Laws of the iterated logarithm