Mathematical Probability II

STAT 582, Spring 2005

Rice University

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

[Outline] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests]

Announcement : 

Time and Place

        According to the official Rice web page.
        Monday Wednesday Friday 10:00-10:50 in (ML 254) check also here


Dr. Rudolf Riedi
Duncan Hall 2082, 713 / 348 3020
Office Hours:  TBA
    or by appointment


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

[Outline] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests]


  • Sidney Resnick "A Probability Path"

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

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


    20%  Homework
    20%  Quiz (Februay 16; on Convergence Concepts)
    30%  First EXAM  (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 Martingales)

    [Outline] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests]

    Classes / Reading

    Past classes and reading assignment will be posted here.

    Topics covered
    reading (Resnick)
    January 12
    no class (examinations)

    January 14 Orientation, Review Stat 581: proba spaces, expectation

    January 17 No Class (Marting Luther King)

    January 19 Review Stat 581: convergence of means

    January 21 VI. Convergence Concepts
    Almost sure, in probability; examples
    Section 6.1 (p167)
    January 24 a.s. implies in probability, Cauchy criterium
    January 26 further relations, Lebesgue's Dominated Convergence
    January 28 Convergence in Lp: def, ex, relation to "a.s." and "in proba"
    Section 6.5, p180-182
    January 31
    Uniform integrability: def, ex, criteria
    February 2
    Unif. integr <=> unif absol contin. and unif 1st moments
    February 4
    Lp convergence <=> conv in proba and unif. integr.
    February 7
    Inequalities. Discuss Homework.

    February 9
    VII.  Laws of Large Numbers. 
    First versions and general WLLN
    February 11
    Sums of indep r.v.:  Levy's theorem
    February 14
    Skorohod's inequality, Kolmogorov's convergence criterium

    February 16 (Wed)
    Quiz on chapter 6 (Convergence concepts)

    February 18
    SLLN: Kronecker's lemma, Records
    February 21
    Kolmogorov's SLLN
    February 23
    Kolmogorov's three series theorem
    VIII. Convergence in Distribution. Basics
    February 25
    Vague and weak convergence, relation to convergence in probabilty
    February 28
    Geometric distr as limit of geometric distr, relation to Poisson process

    March 2
    Scheffe's lemma and convergence in total variation 252-255
    March 4
    Review Uniform Absolute Continuity, Homework

    March 7
    Spring Break

    March 9
    Spring Break
    March 11
    Spring Break
    March 14
    Skorohod's theorem
    March 16
    no class

    March 18
    Continuous mapping, Delta Method
    March 21
    Portmanteau theorem and applications
    March 23
    Slutsky's theorem, Converging Together, m-dependent CLT
    March 25
    Convergence to Types, Extreme Value Distributions
    March 28
    Discussion: Homework 6

    March 30
    Discussion: Homework 5, review chapters VII and VIII.

    April 1
    IX. Characteristic Function
    Convolution, Simple Properties
    April 4
    Expansions of char fct, char fct of the Normal distribution
    April 6
    Uniqueness and Continuity of characteristic functions
    April 8
    Spring Recess

    April 11
    Fourier Inversion formula for densities, Prohorov,Tightness
    303; 309-310
    April 13
    Selection Theorem, completing the proof of Continuity, CLT
    April 15
    X. Conditional Expectation: Basics
    April 18
    Conditional Expectation: simple properties
    April 20
    Conditional expectation as L2-projection, L2-martingale convergence
    348-349 + standard refs
    April 22
    Review by TA

    April 25
    More on characteristic functions and CLT:
    derivatives and moments; Lindeberg-Feller CLT, Lyapunov
    315; 319
    April 27
    More on martingales: up-crossing and convergence of positive martingales
    standard refs
    April 29
    No Class

    [Outline] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests]

    (tex-source and solutions restricted to Rice University)
    Homework sheet Due date (in class) Solutions
    Problem Set 1 [pdf] [ps] [tex] February 4
    Solution 1 [pdf] [ps] [tex]
    Problem Set 2 [pdf] [ps] [tex] February 11
    Solution 2 [pdf] [ps] [tex]
    Problem Set 3 [pdf] [ps] [tex] February 25
    Solution 3 [pdf] [ps] [tex]
    Problem Set 4 [pdf] [ps] [tex] March 4
    Solution 4 [pdf] [ps] [tex]
    Problem Set 5 [pdf] [ps] [tex] March 21
    Solution 5 [pdf] [ps] [tex]
    Problem Set 6 [pdf] [ps] [tex] March 25
    Solution 6 [pdf] [ps] [tex]
    Problem Set 7 [pdf] [ps] [tex] April 13
    Solution 7 [pdf] [ps] [tex]
    Problem Set 8 [pdf] [ps] [tex] April 25
    Solution 8 [pdf] [ps] [tex]


    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.

    [Outline] [Textbooks] [Grading] [Reading assignment] [Homework problems and solutions] [Tests]

    February 16, 2005, in class
    lecture notes allowed
    Test 1 (30%) Due: April 4
    Take home, 180 minutes, (open notes)
    Test 2 (30%) Due: last day of class
    Take home, 3 hours (open books) [pdf] [ps]

    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 .

    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.
    April 19, 2005.  Dr. Rudolf Riedi