Measure Theoretic Probability 2

By Prof. Suprio Bhar | IIT Kanpur

Learners enrolled: 177 | Exam registration: 6

ABOUT THE COURSE:

This course is aimed at the students who have already learnt about basic Probability distributions and Random variables, and are interested in learning about the Mathematical formulation of Probability. This is a follow-up of the NPTEL course “Measure Theoretic Probability 1”. Knowledge of measure theoretic integration will be assumed. PG/Ph.D students and senior UG students are welcome. We shall first discuss the various notions of convergence of random variables and and then focus on the Law of Large Numbers and the Central Limit Theorem.

INTENDED AUDIENCE: This is a follow-up of the NPTEL course “Measure Theoretic Probability 1”. Target audience includes students who have already learnt about basic Probability distributions and Random variables, and are interested in learning the Mathematical formulation of Probability. Knowledge of measure theoretic integration will be assumed. PG/Ph.D students and senior UG students are welcome.

PREREQUISITES: A good background of Real Analysis, Basic Probability Theory (covering Probability distributions and standard Random variables) and Measure Theoretic Integration.

INDUSTRY SUPPORT: This is a course focused on the Mathematical foundations of Probability and not on applications. However, this course is a useful prerequisite towards advanced courses such as Stochastic Calculus and Financial Mathematics. As such, most industries should recognize this course.

Summary

Course Status :	Ongoing
Course Type :	Elective
Duration :	8 weeks
Category :	Mathematics
Credit Points :	2
Level :	Undergraduate/Postgraduate
Start Date :	22 Jul 2024
End Date :	13 Sep 2024
Enrollment Ends :	05 Aug 2024
Exam Registration Ends :	16 Aug 2024
Exam Date :	21 Sep 2024 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1:

Introduction to the course: A review of basic Probability and Measure Theoretic integration
Lp spaces (definition, properties as a Banach space, dual space)

Week 2:

Independence of Events and Random variables, Borel-Cantelli Lemma (second half)
Product measures (construction, integration, Fubini-Tonelli Theorem)
Almost sure convergence of sequences of Random variables (definition and examples)

Week 3:

Other modes of convergence of sequences of Random variables (convergence in probability, convergence in p-th mean, definition and examples)
Relations between various modes of convergence (examples and counter-examples)

Week 4:

Properties of various modes of convergence
Almost sure convergence of series of Random variables (Kolmogorov’s inequality)

Week 5:

Almost sure convergence of series of Random variables – continued (Kolmogorov’s Three series Theorem)
Law of Large numbers (Khinchin’s Weak law, Kolmogorov’s Strong law, applications)
Characteristic Functions (properties, Inversion formulae)

Week 6:

Weak convergence or convergence in distribution (definition and examples)
Equivalent conditions or formulations of weak convergence (Helly-Bray Theorem, Portmanteau Lemma, Levy’s Continuity Theorem)

Week 7:

Equivalent conditions or formulations of weak convergence – continued
Central Limit Theorem (Lindeberg-Levy CLT)

Week 8:

Central Limit Theorems (Lindeberg-Feller CLT, Lyapunov CLT)
Applications
Slutky’s Theorem, Delta Method, Comments on Glivenko-Cantelli Theorem and Berry-Esseen Theorem
Conclusion of the course

Books and references

Probability & Measure Theory (2nd Edition), Robert B. Ash, with contributions from Catherine A. Doleans-Dade. Elsevier.
A Course in Probability Theory (3rd Edition), Kai Lai Chung. Academic Press (Elsevier).
Probability and Measure (3rd Edition), Patrick Billingsley. Wiley.
Probability: Theory and Examples (4th Edition), Rick Durrett. Cambridge University Press.
Probability Essentials (2nd Edition), Jean Jacod and Philip Protter. Springer.

Instructor bio

Prof. Suprio Bhar

IIT Kanpur

Assistant Professor, Department of Mathematics and Statistics, Indian Institute of Technology Kanpur; Ph.D (Indian Statistical Institute, 2015); Research Interests: Stochastic PDEs

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 21 September 2024 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 6 assignments out of the total 8 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kanpur .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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