Mathematical Probability II

STAT 582, Spring 2006

Rice University

This course is the continuation of STAT 581.
The course sequence covers the measure-theoretic foundations of probability.
Open to qualified undergraduates. 

[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]

Last update June 24, 2006

Time and Place

Problem sessions/ make up classes:
check also here for location

Dr. Rudolf Riedi
Duncan Hall 2082, 713 / 348 3020
Office Hours:  Monday 3-4 pm
    or by appointment

[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]


Convergence Concepts and Comparisons
    Almost sure
    In probability
    Lp convergence
Laws of Large Numbers
    Sums of Independent Random Variables
    Weak and Strong versions of the LLN
Convergence in Distribution: Concepts and Comparisons
    Weak and vague convergence.
    Scheffe's lemma: convergence in total variation
    Skorohod: almost sure convergence; Delta Method
    Characteristic functions and Central Limit Theorem
    Conditional Expectation and the Radon-Nykodim Theorem
    Upcrossings and convergence of martingales

[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]


The course will closely follow this book; it is available at the campus bookstore.

Standard references and further suggested reading on Probability Theory
[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]


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]

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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
January 23 Cauchy criterium i.p.
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.
February 1 II
Inequalities. Review a.s versus i.p.
February 3 (for 3/20)
VII.  Laws of Large Numbers. 
First versions and general WLLN
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
February 13
Skorohod's inequality, Kolmogorov's convergence criterium

February 15 I
SLLN: Kronecker's lemma, Records
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

March 6 proof Scheffe's lemma
March 8 I
Skorohod's theorem
March 8 II
Continuous mapping, Delta Method
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
March 27
Exam 1 due

March 27
Portmanteau theorem and applications
March 29 I
Slutsky's theorem
March 29 II
Convergence to Types
April 3
Extreme Value Distributions
April 5 I
Discussion Exam 1

April 5 II
IX. Characteristic Function
Convolution, Simple Properties
April 10
Expansions of char fct, char fct of the Normal distribution
April 12 I
Uniqueness and Continuity of characteristic functions
April 12 II
April 14 (for 4/26)
X. Conditional Expectation: Basics
April 17
Conditional Expectation: simple properties
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

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]

(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]

  [Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]

Late Homework Policy

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]


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)

Honor System

Homework are "open-discussion". This means the following:
Collaboration for homework is encouraged. Any source of information is admissible. However, each student hands in her/his own homework which expresses his/her own understanding of the solution. Simple copying from others does not qualify as "collaboration".

The term "open notes" means that look-up in any self-compiled hand-written source of knowledge is permitted.
The term "open books" means that look-up in any passive source of knowledge is permitted.
No help of any kind is allowed. For instance, no communication is admissible which involves any intelligent entity ---human or artifical--- or any active source which is able to respond to questions .

[Syllabus] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests][Knowledge Milestones]

Knowledge Milestones aquired in this course

Any student with a documented disability needing academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain confidential. Students with disabilities should also contact Disabled Student Services in the Ley Student Center.
February 17, 2005. Dr. Rudolf Riedi