Advanced Probability

General

• Course description

   

• The lecture notes are available in the detailed description of the course below.

   

• Grading:

   

  Four graded homeworks: 5% each
  Midterm: 20%
  Exam: 60%
  Final grade = 60% exam + 20% midterm + 20% graded homeworks

   

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  Staff

  Teacher	E-mail	Voice	Office	Office Hours
  Olivier Lévêque, IC-LTHI	olivier.leveque#epfl.ch	021 693 81 12	INR 132	by appointment
  TA	E-mail	Voice	Office	Office Hours
  ?, ?	?	?	?	?

   

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  Schedule (to be confirmed)

  Type	Day	Hour	Room
  Lectures	Wednesday	2:15 AM – 4:00 PM	INM 203
  Lectures	Thursday	2:15 AM – 3:00 PM	INM 203
  Exercise Sessions	Thursday	3:15 PM – 5:00 PM	INM 203

   

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  Detailed program and lecture notes

  Date	Subject
  Week 1	sigma-fields, random variables
  Week 2	probability measures, distributions, independence
  Week 3	expectation, basic properties, inequalities
  Week 4	convergence of random variables, Borel-Cantelli lemma
  Week 5	laws of large numbers, empirical distribution, normal numbers, Kolmogorov’s 0-1 law
  Week 6	characteristic function, convergence in distribution, Lévy’s continuity theorem, central limit theorem
  Week 7	concentration inequalities: Hoeffding’s inequality, large deviations
  Week 8	moments, Carleman’s theorem + midterm
  Week 9	conditional expectation: definition, properties
  Week 10	martingales: definition, properties, filtrations, stopping times, optional sampling theorem
  Week 11	martingales: martingale transforms, Doob’s decomposition theorem, martingale convergence theorem (I)
  Week 12	martingales: Doob’s maximal inequality, martingale convergence theorem (II), Azuma’s inequality
  Week 13	Poisson process and renewal theory
  Week 14	Renewal theory (cont’d), course revision and more

     

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  Homeworks

  Problem sets	Date	Due	Solutions
  Matlab Homework 1 (graded)	Sept 17	Monday, Oct 5	Matlab Solutions 1 Code
  Homework 1	Sept 17	Sept 24	Solutions 1
  Homework 2	Sept 24	Oct 1	Solutions 2
  Homework 3	Oct 1	Oct 8	Solutions 3
  Matlab Homework 2 (graded)	Oct 8	Monday, Oct 26 extended to Thursday, Oct 29	Matlab Solutions 2 Code
  Homework 4	Oct 8	Oct 15	Solutions 4
  Homework 5	Oct 15	Oct 22	Solutions 5
  Homework 6	Oct 22	Oct 29	Solutions 6
  Homework 7	Oct 29	Nov 5	Solutions 7
  Midterm Exam (replacing Homework 8)	Nov 5, 2:15 PM	Nov 5, 5:00 PM	Midterm Solutions
  Homework 9	Nov 12	Nov 19	Solutions 9
  Matlab Homework 3 (graded)	Nov 12	Monday, Nov 23	Matlab Solutions 3 Code
  Homework 10	Nov 19	Nov 26	Solutions 10
  Homework 11	Nov 26	Dec 3	Solutions 11
  Homework 12	Dec 3	Dec 10	Solutions 12
  Matlab Homework 4 (graded)	Dec 3	Monday, Dec 14	Matlab Solutions 4 Code
  Homework 13	Dec 10	Dec 17	Solutions 13
  Final Exam (2014-2015 version)	Sat, Jan 16, 8:15 AM	Sat, Jan 16, 11:15 AM	Final Solutions

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  Reference textbooks for the course

• Sheldon M. Ross, Erol A. Pekoz, A Second Course in Probability, 1^st edition, www.ProbabilityBookstore.com, 2007.
• Jeffrey S. Rosenthal, A First Look at Rigorous Probability Theory, 2^nd edition, World Scientific, 2006.
• Geoffrey R. Grimmett, David R. Stirzaker, Probability and Random Processes, 3^rd edition, Oxford University Press, 2001.
• More advanced: Richard Durrett, Probability: Theory and Examples, 4^th edition, Cambridge University Press, 2010.
• More advanced: Patrick Billingsley, Probability and Measure, 3^rd edition, Wiley, 1995.