• M 10-11 HZ 118
  
• W 10-11 HZ 118
• W 11-12 HZ 118

• F 11-12 HZ 118 (if announced)
  

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

Syllabus

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 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]

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	p169-172
January 23	Cauchy criterium i.p.	p173-178
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.	p190-194
February 1 II	Inequalities. Review a.s versus i.p.	p186-189
February 3 (for 3/20)	VII.  Laws of Large Numbers.  First versions and general WLLN	p204-208
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	p209-213
February 13	Skorohod's inequality, Kolmogorov's convergence criterium
February 15 I	SLLN: Kronecker's lemma, Records	214-216
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	252-255
March 6	proof Scheffe's lemma	252-255
March 8 I	Skorohod's theorem	258-260
March 8 II	Continuous mapping, Delta Method	261-263
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	263-267
March 27	Exam 1 due
March 27	Portmanteau theorem and applications	263-267
March 29 I	Slutsky's theorem	268-271
March 29 II	Convergence to Types	274-279
April 3	Extreme Value Distributions	274-279
April 5 I	Discussion Exam 1
April 5 II	IX. Characteristic Function Convolution, Simple Properties	293-297
April 10	Expansions of char fct, char fct of the Normal distribution	297-301
April 12 I	Uniqueness and Continuity of characteristic functions	302-305
April 12 II	CLT	312-314
April 14 (for 4/26)	X. Conditional Expectation: Basics	339-342
April 17	Conditional Expectation: simple properties	344-347
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	Review
April 26	No Class, made up for on 4/21
April 27, Thursday	last day of class, exam 2 due

Homework
(tex-source and solutions restricted to Rice University)
 

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
 

Knowledge Milestones aquired in this course

• Convergence Concepts and Comparisons
  •     Be able to formulate and verify in simple cases
    
    • Almost sure Convergence 
    • Convergence in probability
    • Lp convergence
    • Uniform integrability
    • Cauchy criterium for all convergence concepts
  •     Know the relations between these concepts
    • order the types of convergence in strength (e.g.: Lp => Lq => i.p. for p>q>0)
      
    • equivalence in certain special cases
      
      • monotone sequences (with proof): a.s. <=> i.P.
      • sums of indep. r.v.: a.s. <=> i.P.
  • Be able to apply the Borel-Cantelli Lemmas
    
    • to prove or disprove almost sure convergence
  • uniform integrability
    
    • Know sufficient and/or necessary conditions
      
      • sufficient: Crystal ball, domination
      • necessary: unif. bounded first abs. moments (with proof)
      • equivalent: unif. abs. cont. + unif bounded 1st abs. moments (without proof)
        
    • for Lp convergence
      
      • equivalent: conv i.P.  + |X_n|^p is u.i.
        
    • compute criteria for uniform integrability for simple families
      
      • Gaussian
      • Uniform
      • Poisson
      • Exponential
      • Geometric
  • Know the statement and proof and be able to apply:
    
    • the first continuity theorem for a.s. conv and for conv. i. P.
  • Know the statement of the Schwartz inequality
    

• Laws of Large Numbers
  • WLLN: Know the statement of
    
    • L2-version (with proof)
      
    • Khinchin
    • Know the two conditions which are sufficient and necessary
      
  • SLLN: Know the sufficient and necessary condition for the SLLN

• Sums of Independent Random Variables
  • Know the statements of theorems
    
    • Levy's theorem
      
    • Kolmogorov's convergence criterium (with proof, you can use Levy's thm without proving it)
      
    • Kolmogorov's three series theorem
    • Kolmogorov's two series theorem for positive r.v.
      
  • Be able to apply these theorems for simple familiies of r.v.
    • Gaussian
    • Uniform
    • Poisson
    • Exponential
    • Geometric
    • etc

• Convergence in Distribution: Concepts and Comparisons
  •  Be able to formulate and verify in simple cases
    • Vague convergence
    • Weak convergence
    • Proper convergence
    • Convergence in total variation
  • Know the relation between these concepts
    • order them by strength
      
    • know equivalence in special cases
      
      • e.g. vague<=>weak for positive r.v. (with proof)
        
      • e.g. vague<=>weak for proper limits (with proof)
  • ...and the relation to convergence i.P. 
    
    • convergence i.P. implies conv. in distribution (with proof)
    • conv. in distribution to a constant implies conv i.P. (with proof)
  • Know simple facts:
    • whether such limits are unique
    • Distribution functions are determined uniquely on dense set
    • the points of dis-continuity of d.f. are countable (with proof)
  • Scheffe's lemma on conv in total variation 
    • Know the statement
    • be able to verify conv in total variation in simple cases using Scheffe

• Convergence in Distribution: Advanced material
  • Know the following statements and be able to apply in simple situations
    • Skorohod's theorem
    • second continuity theorem for conv. in distr. (with proof)
    • Portmanteau theorem (with proof)
    • Delta method
    • Slutsky's theorem
    • Convergence to types
    • LIndeberg's Central Limit Theorem
  • Know simple properties of characteristic functions
    • boundedness, continuity, etc (with proof)
    • ...in the context of sums of independent r.v. and convolution
    • ...in the context of types of distributions
      
    • Uniqueness and Continuity theorems for Characteristic functions
    • Relation to moments
    • Explicit form for simple distributions such as Normal, exponential, Poisson
      
  • Know the three basic forms of extreme value distributions
  • Central Limit Theorem
    
    • Know the statement for i.i.d. variables (with proof using Uniqueness and Continuity of char fct))
    • Know the Lindeberg-Feller sufficient condition for independent r.v.
      

• Martingales
  • Know the two defining properties of Conditional Expectation
  • Know the simple properies of Conditional Expectation in the case E[ X | Y,Z ]
    
    • Additivity in X
      
    • special cases:
      
      • X,Y,Z mutually independent: E[ X | Y,Z] = E[X]
      • E[ g(Y,Z)*W | Y,Z ] = g(Y,Z) E[ W | Y,Z]
    • Iterated conditional expectation E[ E[ U | V,W ] | V ] = E[ U | V ]
      
    • Interpretation as a projection in the L2-case
      
  • Be able to compute conditional expectations in simple cases
    • using simple properties
    • by explicit computation via conditional probabilities
      
    • ...for discrete and for jointly continuous r.v.
  • Martingales
    • Know the definition of a martingale
      
    • Know the statement of the Martingale-convergence-theorem