Rice University
This course is the continuation of STAT 581.
The course sequence covers the measure-theoretic foundations of
probability.
Open to qualified undergraduates.
Dr. Rudolf Riedi
Duncan Hall 2082, 713 / 348 3020
Office Hours: Monday 3-4 pm
or by appointment
- Sidney Resnick "A Probability Path"
The course will closely follow this book; it is available at the campus bookstore.
20% Homework
20% Quiz (Februay 16; on Convergence Concepts)
30% First EXAM (March 27: on sums of independent r.v. and basics on convergence in distribution)
30% Second EXAM (April: on Advanced material on Convergence in Distribution and on Martingales)
Details on tested material: [Knowledge Milestones]
[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]
Classes / Reading
Class contents and reading assignment will be posted here.
Check the schedule of 2005 for an idea on the course contents and progression.
Topics covered
reading (Resnick)
January 11 Orientation, Review Stat 581: proba spaces, expectation
January 13 Review Stat 581: convergence properties of expectations, Borel-Cantelli
January 16 No Class (Marting Luther King)
January 18 VI. Convergence Concepts
Almost sure, in probability; examples
Section 6.1 (p167)
January 20 a.s. implies in probability
p169-172
January 23 Cauchy criterium i.p.
p173-178
January 25 I Subsequences in the context of "a.s." and "i.P." convergence
Section 6.5, p179-180
January 25 II Convergence in Lp, relation to "a.s." and "i.P."
Section 6.5, p180-182
January 27 (for 2/6)
Uniform integrability
Section 6.5, p182-184
January 30 Uniform integrability and absolute continuity
Section 6.5, p184-186
February 1 I
Conv Lp <=> conv i.p. & u.i.
p190-194
February 1 II
Inequalities. Review a.s versus i.p.
p186-189
February 3 (for 3/20)
VII. Laws of Large Numbers.
First versions and general WLLN
p204-208
February 6
no class (made up for on Jan 27)
February 8 I
HW set 1; proof WLLN
p 208
February 8 II
Sums of indep r.v.: Levy's theorem
p209-213
February 13
Skorohod's inequality, Kolmogorov's convergence criterium
February 15 I
SLLN: Kronecker's lemma, Records
214-216
February 15 II
Homework discussion
February 17 (Friday)
Quiz on chapter 6 (Convergence concepts); Problem session
February 20 Kolmogorov's SLLN 219-222 February 22 I Kolmogorov's three series theorem 226-230 February 22 II VIII. Convergence in Distribution. Basics 247-248 February 27 Distribution functions on dense sets 248-251 March 1 I Vague and weak convergence, relation to convergence in probability 248-251 March 1 II moved to Friday March 3
March 3 (for 3/1)
Geometric distr
Scheffe's lemma and convergence in total variation
252-255
March 6 proof Scheffe's lemma
252-255
March 8 I
Skorohod's theorem
258-260
March 8 II
Continuous mapping, Delta Method
261-263
March 13
Spring Break
March 15
Spring Break
March 20
no class, made up for on Feb 3
handout Exam 1
March 22 I
Discussion Practice Test
March 22 II
Set theory: boundary and continuity of indicator
263-267
March 27
Exam 1 due
March 27
Portmanteau theorem and applications
263-267
March 29 I
Slutsky's theorem
268-271
March 29 II
Convergence to Types
274-279
April 3
Extreme Value Distributions
274-279
April 5 I
Discussion Exam 1
April 5 II
IX. Characteristic Function
Convolution, Simple Properties
293-297
April 10
Expansions of char fct, char fct of the Normal distribution
297-301
April 12 I
Uniqueness and Continuity of characteristic functions
302-305
April 12 II
CLT
312-314
April 14 (for 4/26)
X. Conditional Expectation: Basics
339-342
April 17
Conditional Expectation: simple properties
344-347
April 19 I
Conditional expectation as L2-projection, L2-martingale convergence
348-349 + standard refs
April 19 II
More on martingales: up-crossing and convergence of positive martingales
standard refs
April 19
Hand out exam 2, due last day of class
April 24
Review
April 26
No Class, made up for on 4/21
April 27, Thursday
last day of class, exam 2 due
[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]
Homework
(tex-source and solutions restricted to Rice University)
Homework sheet | Due date (in class) | Solutions |
Problem Set 1 [pdf] [ps] [tex] | February 8 |
Solution 1 [pdf] [ps] [tex] |
Problem Set 2 [pdf] [ps] [tex] | February 13 |
Solution 2 [pdf] [ps] [tex] |
Problem Set 3 [pdf] [ps] [tex] | March 3 |
Solution 3 [pdf] [ps] [tex] |
Problem Set 4 [pdf] [ps] [tex] | March 8 |
Solution 4 [pdf] [ps] [tex] |
Practice Exam Set 5 [pdf] [ps] [tex] | not graded |
Solution 5 [pdf] [ps] [tex] |
Problem Set 6 [pdf] [ps] [tex] | April 10 |
Solution 6 posted April 10 [pdf] [ps] [tex] |
Problem Set 7 [pdf] [ps] [tex] | April 17 |
Solution 7 posted April 17 [pdf] [ps] [tex] |
Problem Set 8 [pdf] [ps] [tex] | not graded |
Solution 8 posted April 21 [pdf] [ps] [tex] |
Homework is due at the beginning of class on the due date. After the due date, but before solutions are handed out, homework can be turned in for 50% credit. In this case, please slip your homework under the instructors's office door, or bring it to class. After solutions are handed out, 0% credit will be issued. You are encouraged to work in groups for homeworks but you will hand in your own solution which you are expected to understand.
[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]
Quiz |
February 17, 2006, in class |
lecture notes allowed |
Test 1 (30%) [pdf] [ps] | Due: March 27 |
Take home, 150+30 minutes, (open notes) |
Test 2 (30%) [pdf] | Due: last day of class |
Take home, 3 hours (open one book +lecture notes) |
Knowledge Milestones aquired in this course