Math 561 Probability I

Renming Song, Spring 2022

• Syllabus

• A Sample Midterm

• Materials on Random Walks

• Lecture Slides

• Final (Due 05/10/22)

• Lecture 1
• Lecture 2
• Lecture of 02/02
• Lecture of 02/04

• Homeworks

• Homework 1: 1.6.2, 1.7.3, 2.1.1, 2.1.9. Due 02/07
• Homework 2: 2.2.1, 2.2.2, 2.2.4, 2.2.5. Due 02/14
• Homework 3: 2.3.15, 2.3.18, 2.4.1, 2.4.2. Due 02/21
• Homework 4: 2.5.1, 2.5.2, 2.5.9, 2.5.10. Due 02/28
• Homework 5: 3.2.6, 3.2.12, 3.2.13, 3.2.14. Due 03/23
• Homework 6: 3.3.13, 3.4.5, 3.4.6, 3.4.12. Due 03/30
• Homework 7: 4.1.8, 4.1.9, 4.1.10, 4.1.11 from the file Materials on Random Walks. Due 04/06
• Homework 8: 4.1.5, 4.1.7, 4.1.8, and this problem Due 04/18
• Homework 9: 4.2.5, 4.2.9, 4.2.10, and this problem Due 04/27
• Homework 10: 4.3.11, 4.4.3, 4.4.6 and 4.4.7 Due 05/02

• Class Diary

• Wednesday, 01/19: sigma-field, measurable space, measure, probability measure, measure space, probability space,measures on the real line, Stieltjes function
  Read: Section 1.1
• Friday, 01/21:measures on high dimensional Euclidean space, random variables, distributions, distribution functions
  Read: Sections 1.1 and 1.2
• Monday, 01/24: Random variables, integration
  Read: Sections 1.3 and 1.4
• Wednesday, 01/26: Integration
  Read: Section 1.4
• Friday, 01/28: Jensen's inequality and Holder inequality
  Read: Section 1.5
• Monday, 01/31: bounded convergence theorem, Fatou's lemma, monotone convergence theorem, expected values
  Read: Sections 1.5 and 1.6
• Wednesday, 02/02: Theorem 1.6.8, change of variable formula, \pi-system, \lambda-system, \pi-\lambda theorem, product measures, Fubini's theorem
  Read: Sections 1.6 and 1.7
• Friday, 02/04:Independence, sums of independent random variables
  Read: Section 2.1
• Monday, 02/07: Kolmogorov's extension theorem, L^2 weak law, polynomial approximation
  Read: Sections 2.1.4 and 2.2.1
• Wednesday, 02/09: Triangular arrays, coupon colelctor's problem, occupancy problem, weak law for triangular arrays, weak law of large numbers
  Read:Section 2.2., 2.2.3
• Friday, 02/11: weak law of large numbers, St.Petersburg paradox, Borel-Cantelli lemma
  Read: Sections 2.2.3, 2.3
• Monday, 02/14: L^2 strong law of large numbers, 2nd Borel-Cantelli lemma,
  Read:Section 2.3
• Wednesday, 02/16: head runs, strong law of large numbers, renewal theory,
  Read: Sections 2.3 and 2.4
• Friday, 02/18: empirical distribution, The Glivenko-Cantelli theorem, tail events, Kolmogorov's 0-1 law, Kolmogorov's maximal; inequality
  Read:Sections 2.4 and 2.5
• Monday, 02/21: Kolmogorov's 3 series theorem, rates of convergence
  Read:Section 2.5
• Wednesday, 02/23: Infinite mean
  Read: Section 2.5.2
• Friday, 02/25: Weak convergence, characterization of weak convergence,
  Read:Sections 3.1, 3.2.1 and 3.2.2
• Monday, 02/28: continuous mapping theorem, Portmanteau theorem, Helly's selection theorem, tightness
  Read:Section 3.2.2
• Wednesday, 03/02:tightness, characteristic functions, properties of characteristic functions, the inversion formula
  Read: Sections 3.2.2, 3.3.1
• Friday, 03/04: the inversion formula, continuity theorem
  Read:Sections 3.3.1 and 3.3.2
• Monday, 03/07: moments and derivatives, Polya's criterion
  Read:Sections 3.3.3 and 3.3.4
• Wednesday, 03/09: Centrla limit theorem
  Read: Section 3.4.1
• Friday, 03/11: Midterm
• Monday, 03/21: Lindeberg-Feller theorem, the converse of the three series theorem
  Read:Section 3.4.2
• Wednesday, 03/23: Infinite variance, rates of convergence
  Read:Section 3.4.2, 3.4.4, 2.5
• Friday, 03/25: permutable events, exchangeable sigma-field, Hewitt-Savage 0-1 law
  Read:Section 2.5 (or Section 4.1 in the file Materials on Random Walks)
• Monday, 03/28:stopping times and their applications
  Read: Section 4.1 in the file Materials on Random Walks.
• Wednesday, 03/30: Wald's equation, simple random walks, Wald's 2nd equation
  Read: Section 4.1 in the file Materials on Random Walks.
• Friday, 04/01: Applications of Wald's 2nd equation to random walks, recurrence, transience, simple symmetric random walks on Z^d
  Read:Sections 4.1 and 4.2 in the file Materials on Random Walks.
• Monday, 04/04: recurrence and transience, simple symmetric random walk on Z^d
  Read:Section 4.2 in the file Materials on Random Walks.
• Wednesday, 04/06: recurrence and transience of general random walks in R^d.
  Read:Section 4.2 in the file Materials on Random Walks.
• Friday, 04/08: recurrence and transience of general random walks in R^d (cont)
  Read:Section 4.2 in the file Materials on Random Walks.
• Monday, 04/11: Trul3 3d random walks are transient. conditional expectation
  Read:Section in the file Materials on Random Walks. Section 4.1 in the book.
• Wednesday, 04/13: Examples and properties of conditional expectations
  Read:Sections 4.1.1 and 4.1.2
• Friday, 04/15: Martingales, submartingales, supermartingales, examples, properties
  Read:Section 4.2
• Monday, 04/18: upcrossing inequality, martingale convergence theorem
  Read:Section 4.2
• Wednesday, 04/20: bounded increments, Doob's decomposition, branching processes
  Read:Sections 4.3.1 and 4.3.4.
• Friday, 04/22: branchung processes, Doob's inequality, L^p maximal inequality
  Read:Sections 4.3.4 and 4.4
• Monday, 04/25: , L^p convergence theorem, applications to branching processes, uniform integrability and L^1 convergence
  Read:Sections 4.4 and 4.6
• Wednesday, 04/27: L^1 convergenc of martingales, dominated convergence theorem for conditional expectations
  Read:Section 4.6
• Friday, 04/29: backwards martingale, application to strong law of large numbers, optional stopping theorem
  Read:Sections 4.7 and 4.8
• Monday, 05/02: optional stopping theorem
  Read:Section 4.8
• Wednesday, 05/04:
  Read:Section