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Date Topics
Text Assignments
1 1/20
Definition of probability
Chapter 1
Assignment due 2/5
Hand in 1 and 3.
Solutions 2 1/22
Algebras and sigma-algebras
Caratheodory's extension Theorem
Chapter 1
3 1/27
Constructing probability spaces:
Coin toss space; probability measures on R
Chapter 1, Section 1 (lectures 1-3)
4 9/15
Independence; product probability spaces
Chapter 1, Sections
1-3
This supercedes the previous post.
Beware of typos. I have not proof read!
Please bring typos to my attention.
5 2/3
Monotone Class theorem and applications
Chapter 1
Problems due 2/12
Hand in 1 and 3
Solutions6 2/5
Random variables
Notes on random variables
7 2/10
Random variables; Basic types.
Coupling random variables.
8 2/12
Expectation; weak law of large numbers
Problems due 2/26
Hand in 5, 7, 11
Solutions9 2/17
Independent random variables
Borel-Cantelli
Kolmogorov 0-1 law
10 2/19
Strong law of large numbers
Lecture Notes, Chapter 3
11 2/24
Kolmogorov's three series theorem
Chapter 3 of lecture notes
12 2/26
Application of three series theorem
Folland, Chapter 1
Chapter 3 of lecture notes
13 3/3
Large deviations, part I
Problems due 3/12
Hand in 1, 4, 6, 1214 3/5
Large deviations, part II
Class Notes
15 3/10
Large Deviations; conclusion
Class Notes (see link above)
16 3/12
Convergence in distribution;
characteristic functions Notes on convergence in
distribution and CLT
17 3/24
Convergence in distribution: Helley-Bray Theorem, Tightness
18 3/26
Characteristic functions and convergence in distribution
Corrected Chapter 4 on CLT
Problem Set 5
19 3/31
Central Limit theorem; statement/proof
Chapter 4
20 4/2
Normal random vectors
21 4/7
Levy's construction of Brownian motion
Notes on Levy's construction
22 4/9
Conditional Expectation
Notes on conditional expectation
23 4/14
Conditional Expectation
24 4/16
Martingales and Markov chains
Notes on Martingales
25 4/21
Martingale Theory
Problem set 6 Hand in 4, 6, 11,
by May 4.