Advanced probability and applications
Weekly outline

"Probability theory is nothing but common sense reduced to calculation." PierreSimon de Laplace, 1812 (see other interesting quotes from PierreSimon de Laplace)(Zoom link in case of online sessions)
Mediaspace channel for the course (please note that these videos were made for a previous version of the course: there will be some differences with this year's version)Recordings of live lectures (starting April 19)Lectures:
 In presence in room CM 1 120 on Wed 24 PM and in room INR 219 on Thu 810 AMExercise sessions:
 in presence in room INR 219 on Thu 1012 AM
Grading scheme:
 graded homeworks 20%
 midterm 20%
 final exam 60%Principle for the graded homeworks: each week, one exercise is starred and worth 2% of the final grade; the best 10 homeworks (out of 12) are considered.
Q&A session: Friday, June 23, 10:1512:00 AM, in room INR 113
Final exam: Wednesday, June 28, 9:1512:15 AM, in room CM 1 105
Course instructor:
 allowed material: two cheat sheets (i.e., four singlesided A4 handwritten pages)
 please note that the content of the exam will focus more on the second part of the course (but also on the first part)
Olivier Lévêque // LTHI // INR 132 // 021 693 81 12 // olivier.leveque@epfl.ch
Teaching assistants:Bora Dogan // LINX // bora.dogan@epfl.chAnand George // SMILS // anand.george@epfl.chLifu Jin // lifu.jin@epfl.chReferences:
Sheldon M. Ross, Erol A. Pekoz, A Second Course in Probability, 1st edition, www.ProbabilityBookstore.com, 2007.
Jeffrey S. Rosenthal, A First Look at Rigorous Probability Theory, 2nd edition, World Scientific, 2006.
Geoffrey R. Grimmett, David R. Stirzaker, Probability and Random Processes, 3rd edition, Oxford University Press, 2001.
Geoffrey R. Grimmett, David R. Stirzaker, One Thousand Exercises in Probability, 1st Edition, Oxford University Press, 2001.
Sheldon M. Ross, Stochastic Processes, 2nd edition, Wiley, 1996.
William Feller, An Introduction to Probability Theory and Its Applications, Vol. 1&2, Wiley, 1950.
(more advanced) Richard Durrett, Probability: Theory and Examples, 4th edition, Cambridge University Press, 2010.(more advanced) Patrick Billingsley, Probability and Measure, 3rd edition, Wiley, 1995.

Wed: Sigmafields (sections 1.1, 1.2, 1.3)
Thu: Random variables (sections 1.4, 1.5), probability measures (section 2.1) 
Wed: Distributions of random variables (sections 2.2, 2.3, 2.4, 2.5)
Thu: Independence (sections 3.1, 3.2, 3.3, 3.4) 
Wed: Do independent random variables really exist? (section 3.5), convolution (section 3.6)
Thu: Expectation (section 4) 
Wed: Characteristic function (section 5), random vectors (section 6.1)
Thu: Gaussian random vectors (sections 6.2, 6.3) 
Wed: Inequalities (section 7)
Thu: Convergences of sequences of random variables and BorelCantelli lemma (sections 8.1, 8.2, 8.3, 8.4)

Wed: Laws of large numbers  weak and strong, proof; convergence of the empirical distribution (sections 8.5, 8.6)
Thu: Kolmogorov's 01 law, St Petersburg's paradox and extension of the weak law (sections 8.7, 8.8)
NB: Because of the midterm next week, Hwk 6 is due on Thursday, April 20 only!

Wed: Convergence in distribution (section 9.1) and vague convergence (appendix A.4)
Thu, 8:1510:00 AM, in room CM 1 4: Midterm
 content: all the course + exercises until week 6 (except hwk 6)
 allowed material: one cheat sheet (i.e., two singlesided A4 handwritten pages) 
Wed: equivalent definitions of convergence in distribution, Lindeberg's principle (sections 9.3, 9.4)
Thu: proofs of the central limit theorem (section 9.4, 9.5), coupon collector problem (section 9.6) 
Wed (no live lecture): Moments and Carleman's theorem (section 10, prerecorded video)Thu: Hoeffding's inequality and large deviations principle (sections 11.1, 11.2)

Wed: free time or revision time!
Thu: Conditional expectation (section 12) 
Wed (no live lecture): Martingales: definition and basic properties, stopping times (sections 13.1, 13.2, prerecorded video 1, prerecorded video 2).Thu: Doob's optional stopping theorem, reflection principle (sections 13.3, 13.4)
Please read also the lecture notes and watch out that in the following, I will consider the general definition given there for martingales, not the one restricted to the squareintegrable case. 
Wed: Martingale transforms and Doob's decomposition theorem (sections 13.5, 13.6)
Brownian motion (a topic not in the lecture notes)Thu: Ascension (no class)In replacement, there will be an exercise session on Monday, May 22, 1012 AM (room INR 113). 
Wed: Martingale convergence theorem (v1) and consequences (sections 14.1, 14.2, 14.3)
Thu: Proof of the theorem (section 14.4) 
Wed: MCT (v2), and generalization to sub and supermartingales (sections 14.5, 14.6)Thu: Azuma's and McDiarmid's inequalities (section 14.7)