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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.

**Assessment**

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.

**Certification**

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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 | ||

Assessment | |||

Submit Your Assignment | 00:00:00 | ||

Certification | 00:00:00 |

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