Announcement : 

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

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

Topics 

Textbook

Sidney Resnick "A Probability Path"

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

• 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'

Grading

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)

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

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]

 

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.

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]