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 :

- Problem sessions are offered Wednesday's 2-3 pm DH 1042

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

Martingales

Conditional Expectation and the Radon-Nykodim Theorem

Upcrossings and convergence of martingales

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

Martingales

Conditional Expectation and the Radon-Nykodim Theorem

Upcrossings and convergence of martingales

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

- A. Papoulis, `Probability, Random Variables, and Stochastic Processes'
- W. Davenport, `Probability and Random Processes'
- W. Feller, `An Introduction to Probability Theory and Its Applications'
- P. Billingsley, `Probability and Measure'

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

p169-172

January 26 further relations, Lebesgue's Dominated Convergence

p173-178

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

p182-184

February 2

Unif. integr <=> unif absol contin. and unif 1st moments

p184-186

February 4

Lp convergence <=> conv in proba and unif. integr.

p190-194

February 7

Inequalities. Discuss Homework.

p186-189

February 9

VII. Laws of Large Numbers.

First versions and general WLLN

p204-208

February 11

Sums of indep r.v.: Levy's theorem

p209-213

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

214-216

February 21

Kolmogorov's SLLN

219-222

February 23

Kolmogorov's three series theorem

VIII. Convergence in Distribution. Basics

226-230

247-248

February 25

Vague and weak convergence, relation to convergence in probabilty

248-251

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

258-260

March 16

no class

March 18

Continuous mapping, Delta Method

261-263

March 21

Portmanteau theorem and applications

263-267

March 23

Slutsky's theorem, Converging Together, m-dependent CLT

268-271

March 25

Convergence to Types, Extreme Value Distributions

274-279

March 28

Discussion: Homework 6

March 30

Discussion: Homework 5, review chapters VII and VIII.

April 1

IX. Characteristic Function

Convolution, Simple Properties

293-297

April 4

Expansions of char fct, char fct of the Normal distribution

297-301

April 6

Uniqueness and Continuity of characteristic functions

302-305

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

307-309;311-312;312-314

April 15

X. Conditional Expectation: Basics

339-342

April 18

Conditional Expectation: simple properties

344-347

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

301-302

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]

**Homework**

(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] |

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]

Quiz |
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:

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".

Tests:

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.