Probability Theory

Academic unit or major

Undergraduate major in Mathematics

Instructor(s)

Masato Hoshino

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

unknown

Class

-

Course Code

MTH.C361

Number of credits

200

Course offered

2025

Offered quarter

4Q

Syllabus updated

Aug 25, 2025

Language

Japanese

Syllabus

Course overview and goals

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.

Course description and aims

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.

Keywords

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

Competencies

• Specialist skills
• Intercultural skills
• Communication skills
• Critical thinking skills
• Practical and/or problem-solving skills

Class flow

Standard lecture course.

Course schedule/Objectives

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

None required.

Reference books, course materials, etc.

David Williams, ``Probability with Martingales'', Cambridge University Press

Evaluation methods and criteria

Final exam.

Related courses

• MTH.C211 ： Applied Analysis I
• MTH.C212 ： Applied Analysis II
• MTH.C305 ： Real Analysis I
• MTH.C306 ： Real Analysis II

Prerequisites

Students are expected to have passed Applied Analysis I, Applied Analysis II, Real Analysis I and Real Analysis II.