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
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course.
Course schedule | Required learning | |
---|---|---|
Class 1 | Probability space, probability measure | Details will be provided during each class session |
Class 2 | Random variable and its distribution, expectation | Details will be provided during each class session |
Class 3 | Change of variable formula for expectation, examples of basic distributions | Details will be provided during each class session |
Class 4 | Independence and its basic properties, joint distribution and marginal distribution | Details will be provided during each class session |
Class 5 | Independence and distributions, independence and expectations, construction of independent, identically distributed random variables | Details will be provided during each class session |
Class 6 | Concepts of convergence of random variables | Details will be provided during each class session |
Class 7 | Examples of convergence of random variables, weak law of large numbers | Details will be provided during each class session |
Class 8 | Borel-Cantelli's lemma, strong law of large numbers | Details will be provided during each class session |
Class 9 | Proof of strong law of large numbers | 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 | Convergence in law of random variables | 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 | Basics of stochastic processes | Details will be provided during each class session |
None required.
J. W. Lamperti, "Probability Theory." 2nd ed., John Wiley & Sons, Inc.
R. Durrett, "Probability: Theory and Examples" 4th ed., Cambridge Univ. Press.
D. Williams, "Probability with Martingales", Cambridge Univ. Press.
Final exam (about 60%) and report (about 40%).
Students are expected to have passed Applied Analysis I, Applied Analysis II, Real Analysis I and Real Analysis II.