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This course covers topics such as sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales. In this course, some topics will also be studied in the classroom and they are probability Spaces and Sigma-Algebras, extension Theorems: A Tool for Constructing Measures, Random Variables and Distributions, Laws of Large Numbers and Independence and Zero-One Laws and Maximal Inequalities.


This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.


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Course Credit: MIT

Course Curriculum

Module: 01
Lecture1 Probability Spaces and Sigma-Algebras 00:30:00
Lecture2 Extension Theorems A Tool for Constructing Measures 00:30:00
Lecture3 Random Variables and Distributions 00:30:00
Lecture4 Integration 00:20:00
Lecture5 More Integration and Expectation 00:30:00
Lecture6 Laws of Large Numbers and Independence 00:30:00
Lecture7 Sums of Random Variables 00:40:00
Module: 02
Lecture8 Weak Laws and Moment-Generating and Characteristic Functions 00:50:00
Lecture9 Borel-Cantelli and the Strong Law of Large Numbers 00:30:00
Lecture10 Zero-One Laws and Maximal Inequalities 00:20:00
Lecture11 Independent Sums and Large Deviations 00:30:00
Lecture12 DeMoivre-Laplace and Weak Convergence 00:30:00
Lecture13 Large Deviations 00:30:00
Lecture14 Weak Convergence and Characteristic Functions 00:30:00
Module: 03
Lecture15 Characteristic Functions and Central Limit Theorem 00:30:00
Lecture16 Central Limit Theorem Variants 00:30:00
Lecture17 Poisson Random Variables 00:40:00
Lecture18 Stable Random Variables, Higher Dimensional Limit Theorems 00:30:00
Lecture19 Infinite Divisibility and Levy Processes 00:15:00
Lecture20 Random Walks 01:10:00
Lecture21 Reflections and Martingales 00:30:00
Module: 04
Lecture22 More on Martingales 00:50:00
Lecture23 More on Martingales (cont.) 00:40:00
Lecture24 Even More on Martingales 00:40:00
Lecture25 Still More Martingales 00:30:00
Lecture26 Markov Chains 00:40:00
Lecture27 More Markov Chains 00:40:00
Lecture28 Additional Material on Markov Chains 00:30:00
Module: 05
Lecture29 Ergodic Theory 00:20:00
Lecture30 More Ergodic Theory 00:30:00
Lecture31 Ergodic Theory 00:20:00
Lecture32 Brownian Motion 00:30:00
Lecture33 More Brownian Motion 00:30:00
Lecture34 Even More Brownian Motion 00:40:00
Lecture35 Last Lecture 00:30:00
Submit Your Assignment 00:00:00
Certification 00:00:00

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