Modern Probability Theory Basics

Author: Wang Jiagang
Publisher:
Publish Date: 2005-08-01
Features: The progress of human civilization and social development is constantly benefited from and influenced by mathematics. The application and development of mathematical science have firmly established its foundational role in all scientific and technological fields as well as many humanities. In the current era, mathematics is breaking through traditional application boundaries and permeating almost every field of human knowledge. Its interaction with other disciplines has become unprecedentedly active, increasingly contributing directly to human material production and daily life, and serving as a key to unlocking numerous opportunities for its practitioners.
Probability theory is a branch of mathematics that studies the quantitative laws of random phenomena. The modern development of probability theory, which flourished in the 1930s, originated from the establishment of its logical foundation and the practical needs of science, technology, and social practice. Today, probability theory not only itself—such as in the fields of stochastic processes, stochastic analysis, and limit theory—has garnered widespread attention, but it is also closely linked to the development of disciplines such as mathematical statistics, mathematical finance, and biomathematics.
This book starts with sets and basic analysis, using the perspective and methods of measure theory to systematically discuss fundamental concepts, common tools, and methods in probability theory, such as events, probability, random variables, and expectations. Building on this foundation, it introduces key results and methods in areas like independent sequences of random variables, conditional expectations, and martingale sequences, thereby providing readers with the necessary foundation and training for further study in probability theory, mathematical statistics, and related disciplines.
Using measure theory as a tool, this book systematically elaborates on the fundamental concepts of probability theory (such as events, random variables, probability, and expectations) and also introduces major results in areas like independent sequences of random variables, conditional expectations, and martingale sequences, thus providing readers with the necessary foundation for in-depth study in modern probability theory, stochastic processes, and mathematical statistics. This book can serve as a textbook or teaching reference for undergraduate and graduate students, as well as for students, teachers, and researchers in related fields.