Mathematical Probability II

STAT 582, Spring 2007

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]

Announcement (Last update April 16, 2007)

Test 2 due Wednesday, April 25 (30%) [pdf]

Time and Place

Problem sessions and 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 (mid Februay; on Convergence Concepts)
30%  First EXAM  (end March: 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 2006 for an idea on the course contents and progression.

Topics covered
reading (Resnick)
January 8 no class

January 10 Orientation, Review Stat 581: proba spaces, expectation
January 12 Review Stat 581: convergence properties of expectations, Borel-Cantelli

January 16 No Class (Marting Luther King)

January 17 cancelled due to severe weather

January 19 I VI. Convergence Concepts
Almost sure, in probability; examples
Section 6.1 (p167)
January 19 II (Make-up for January 17) a.s. implies in probability, Cauchy criterium i.p.
January 22 Cauchy i.P. implies a subsequence converges a.s.
January 24 Subsequences ("a.s." and "i.P."), 1st Continuity thm, Lebesgue DCT
p174-178, p175
January 26 [9-10] Convergence in Lp, relation to "a.s." and "i.P."
Section 6.5, p180-182
January 26 [10-11]
Uniform integrability
Section 6.5, p182-184
January 29 Uniform integrability and moment conditions
Section 6.5, p184-186
January 31
Conv L1 <=> conv i.p. & u.i.
February 2, [9-10]
Absolute Continuity & bounded 1st moments <=> u.i.

February 2, [10-11]
Conv Lp <=> conv i.P. & |X_n|^p is u.i.
Inequalities. p186-189
February 5
VII.  LLN. Tail equivalence and L2 version of WLLN
February 7
simple sufficient and the exact equivalent conditions of the WLLN
p 205-208
February 9
Sums of indep r.v.:  Levy's theorem
February 12, [9:15]
Quiz on chapter 6 (Convergence concepts)

February 12, [10-11]
Skorohod's inequality, Kolmogorov's convergence criterium

February 14
Kronecker's lemma, Records
February 16 [9:30 - 10:50]
Kolmogorov's SLLN 219-222
February 19 Kolmogorov's three series theorem 226-230
February 21 VIII. Convergence in Distribution. Basics, Dense sets 247-248, 248-251
February 23 Vague and weak convergence, relation to convergence in probability 248-251
February 26
Geometric and exponential distr
Scheffe's lemma and convergence in total variation

Feb 28 Skorohod's theorem
March 2 [9:30-11]
Continuous mapping, Delta Method
March 5
Spring Break

March 7
Spring Break

March 9
Spring Break
March 12
no class, made up for on Feb 16 and March 2
March 14
March 16
Slutsky's theorem
March 19
Convergence to Types
March 21
Extreme Value Distributions
March 23
Extreme Value Distr. (proof)

March 26
IX. Characteristic Function
Simple Properties
March 28
Expansions of char fct, char fct of the Normal distribution
March 30
Uniqueness of characteristic functions
April 2
Continuity of char fct and CLT
April 4
X. Conditional Expectation: Basics
April 6
April 9
Conditional Expectation: simple properties
April 11
Advanced properties
348-349 + standard refs
April 13
More on martingales: up-crossing and convergence of positive martingales
standard refs
April 16
Hand out exam 2, due last day of class

April 16
Conditional Exp for continuous r.v.

April 18
L2-martingale convergence

April 20

April 23
No Class, made up for on 1/26

April 25
no class, made up for on 2/2
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] Jan 31, 2007 posted Feb 5 [pdf] [ps] [tex]
Problem Set 2 [pdf] [ps] [tex] Feb 7, 2007
posted Feb 7 [pdf] [ps] [tex]
Problem Set 3 [pdf] [ps] [tex] Feb 19, 2007
posted Feb 19 [pdf] [ps] [tex]
Problem Set 4 [pdf] [ps] [tex] Feb 28, 2007
posted March 1 [pdf] [ps] [tex]
Practice Exam Set 5 [pdf] [ps] [tex] March 16, 2007
posted March 6 [pdf] [ps] [tex]
Problem Set 6 [pdf] [ps] [tex] March 28
posted April 4 [pdf] [ps] [tex]
Problem Set 7 [pdf] [ps] [tex] April 4
posted April 11 [pdf] [ps] [tex]
Problem Set 8 Practice Test 2[pdf] [ps] [tex] due April 11
Solution 8 posted April 15 [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 12, in class 9:15 lecture notes allowed
Test 1 (30%) [pdf] Due: March 23
Take home, 150 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.
December 12, 2006. Dr. Rudolf Riedi