In this course, we introduce fundamental concepts in measure-theoretic probability theory, and we study basic limit theorems by means of those concepts. We first define several basic concepts which form a basis of the whole probability theory, and study their elementary properties. More precisely, we introduce probability space, probability measure, random variables, probability distribution, expectation and independence. On the basis of these preparations, we formulate and prove the law of large numbers and the central limit theorem, which are ones of most fundamental limit theorems.
Kolmogorov's axiomization of probability theory by means of measure theory provide a rigorous mathematical basis to the concept of probability, while it had been broadly used in the literature even before. In particular, this "revolution" makes it possible to develop arguments involving infinity precisely and we can state several limit theorems mean without ambiguity. Through this course, we will reveal how we justify concepts, theorems and computations, which were treated intuitively, and what properties they enjoy.
Students are expected to:
Be able to follow arguments of measure-theoretic probability theory.
Be able to compute characteristics (expectation, variance and characteristic function etc.) of elementary distributions.
Understand the definition and properties of convergences of random variables and distributions, and be able to explain elementary examples.
Be able to explain how we formulate the law of large numbers and the central limit theorem rigorously.
Be able to explain an outline of the proof of these limit theorems.
Probability space, probability measure, random variable, probability distribution, expectation, independence, almost-sure convergence, convergence in probability, Borel-Cantelli's lemma, law of large numbers, convergence in law, characteristic function, central limit theorem,martingale
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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Standard lecture course.
Course schedule | Required learning | |
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Class 1 | Probability space, probability measure, Borel-Cantelli theorem | Details will be provided during each class session |
Class 2 | Random variables, Independence | Details will be provided during each class session |
Class 3 | Kolmogorov's 0-1 law | Details will be provided during each class session |
Class 4 | Expectation | Details will be provided during each class session |
Class 5 | Conditional Expectations | Details will be provided during each class session |
Class 6 | Discrete Time Martingale | Details will be provided during each class session |
Class 7 | Optional Stopping Theorem, Martingale Convergence Theorem | Details will be provided during each class session |
Class 8 | strong law of large numbers | Details will be provided during each class session |
Class 9 | Characteristic Functions | Details will be provided during each class session |
Class 10 | Applications of strong law of large numbes， convergence of probability measures | Details will be provided during each class session |
Class 11 | Weak Convergence | Details will be provided during each class session |
Class 12 | Basic properties of characteristic functions, examples of characteristic functions | Details will be provided during each class session |
Class 13 | Characteristic functions and distributions | Details will be provided during each class session |
Class 14 | Central limit theorem | Details will be provided during each class session |
Class 15 | Some additional topics | Details will be provided during each class session |
None required.
David Williams, ``Probability with Martingales'', Cambridge University Press
Final exam (about 50%) and report (about 50%).
Students are expected to have passed Applied Analysis I, Applied Analysis II, Real Analysis I and Real Analysis II.