Measure Theoretic Probability 2

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 useful prerequisite towards advanced courses such as Stochastic Calculus and Financial Mathematics. As such, most industries should recognize this course.

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