Math 561 Probability I

Renming Song, Spring 2024

• Syllabus

• A Sample Midterm

• Final Due 05/06

• Materials on random walks

• Homeworks

• Homework 1: 1.7.3, 2.1.1, 2.1.4, 2.1.9 Due 02/09
• Homework 2: 2.2.1, 2.2.2, 2.2.4, 2.2.5 Due 02/16
• Homework 3: 2.3.15, 2.3.18, 2.4.1, 2.4.2 Due 02/23
• Homework 4: 2.5.1, 2.5.2, 2.5.9, 2.5.10 Due 03/04
• Homework 5: 3.2.6, 3.2.12, 3.2.13, 3.2.14 Due 03/22
• Homework 6: 3.3.13, 3.4.5, 3.4.6, 3.4.12 Due 03/29
• Homework 7: 4.1.8, 4.1.9, 4.1.10, 4.1.11 from the file Materials on Random Walks. Due 04/05
• Homework 8: 4.1.5, 4.1.7, 4.1.8, and This problem Due 04/17
• Homework 9: 4.2.5, This problem, 4.4.6, 4.4.7 Due 04/26

• Class Diary

• Wednesday, 01/17: sigma-fields, measure, probability, Borel sigma-fields, Stieltjes functions, measures on the real line
  Read: Section 1.1
• Friday, 01/19: measures on d-dimensional Eucldean spaces, random variables, distributions, distribution functions
  Read: Sections 1.1 and 1.2
• Monday, 01/22: random variables, integration
  Read: Sections 1.3 and 1.4
• Wednesday, 01/24: integration
  Read: Section 1.4
• Friday, 01/26: Properties of integrals
  Read: Section 1.5
• Monday, 01/29: properties of integrals, expected value
  Read: Sections 1.5 and 1.6
• Wednesday, 01/31: Product measures, Fubini's theorem
  Read: Section 1.7
• Friday, 02/02: Independence
  Read: Section 2.1
• Monday, 02/05: Sums of independent random variables constructing independent random variables
  Read: Section 2.1
• Wednesday, 02/07: Weak law of large numbers
  Read: Section 2.2
• Friday, 02/09: Weak law of large numbers
  Read: Section 2.2
• Monday, 02/12: Borel Cantelli lemmas
  Read: Section 2.3
• Wednesday, 02/14: Borel Cantelli lemmas
  Read: Section 2.3
• Friday, 02/16: Strong law of large numbers
  Read: Section 2.4
• Monday, 02/19: Tail sigma-field, Kolmogorov's 0-1 law, Kolmogorov's maximal inequality, Kolmogorov's 3 series theorem
  Read: Section 2.5
• Wednesday, 02/21: proof of SLLN using the 3 series theorem,rates of convergence
  Read: Section 2.5
• Friday, 02/23: infinite mean
  Read: Section 2.5.2
• Monday, 02/26: weak convergence
  Read: Section 3.2
• Wednesday, 02/28: Helly's selecion theorem, tightness.
  Read: Section 3.2
• Friday, 03/01: characteristic fuctions, inversion formula
  Read: Section 3.3
• Monday, 03/04:Inversion formula, continuity theorem
  Read: Section 3.3
• Wednesday, 03/06: moments and derivatives of characteristic functions, central limit theorem for iid sequences
  Read: Sections 3.3 and 3.4
• Friday, 03/08: Test. Median: 75. Score Distribution: 95--99 (3), 90--94 (2); 85--89 (2); 80--84 (2); 75--79 (3); 70--74 (2); 65--69 (1); 60--64 (5); --50 (3)
• Monday, 03/18: The Lindeberg-Feller theorem, the converse of the three series theorem
  Read: Section 3.4.2
• Wednesday, 03/20: Example 3.4.13, exchangeable sigma-field,
  Read: Section 3.4.1, and the materials on random walks
• Friday, 03/22: Hewitt-Savage 0-1 law and its application to random walk on the real line, stopping times.
  Read: Materials on random walks
• Monday, 03/25: Theorem 4.1.3, ship operator, interates of stopping times, Theorem 4.1.4
  Read: Materials on random walks
• Wednesday, 03/27: Wald's identities and application to random walks.
  Read: Materials on random walks
• Friday, 03/29: Theorem 4.1.7, Lemma 4.1.8, recurrent points, possible points, Theorem 4.2.1
  Read: Materials on random walks
• Monday, 04/01: Theorem 4.2.3, Lemma 4.2.4, Lemma 4.2.5, Theorem 4.2.6
  Read: Materials on random walks
• Wednesday, 04/03: Theorem 4.2.7, Theorem 4.2.8, Theorem 4.2.9 and Theorem 4.2.10
  Read: Materials on random walks
• Friday, 04/05: Theorem 4.2.10, Theorem 4.2.13 from the materials on random walks, conditional expectation
  Read: Materials on random walks and Section 4.1
• Monday, 04/08: Properties of conditinal expectations
  Read: Sections 4.1.1 and 4.1.2
• Wednesday, 04/10: Properties of conditinal expectations, martingales, submartingales and supermartingales
  Read: Sections 4.1.2 and 4.2
• Friday, 04/12: upcrossing inequality
  Read: Section 4.2
• Monday, 04/15: martingale convergence theorem, martingales with bounded increments
  Read: Sections 4.2 and 4.3.1
• Wednesday, 04/17: branching processes, Theorem 4.4.1
  Read: Section 4.3.4, Theorem 4.4.1
• Friday, 04/19: Doob's inequality, convergence in L^p, p>1.
  Read: Section 4.4
• Monday, 04/22: Uniform integrability, convergence in L^1
  Read: Section 4.6
• Wednesday, 04/24: Dominated convergence theorem for conditional expectations, backwards martingales
  Read: Sections 4.6 and 4.7
• Monday, 04/29:
  Read: Section
• Wednesday, 05/01:
  Read: