[Outline of the book]
Statistics is an interdisciplinary field, and it is particularly connected to the probability theory. The introduction of new methodologies from various fields of mathematics, including the probability theory, has often been a major catalyst for the development of the statistics theory. The martingale is one of the most important objects in the probability theory. Examples of its great contributions to statistics include the counting process approach to survival analysis and diffusion process-based modeling in mathematical finance.
Dr. Nishiyama deals with two themes related to martingales in this book.
First, he summarizes the basics of martingales and explains the proof of the martingale convergence theorem in a way that is different from the original approach by J.L. Doob. In Chapter 2, some questions are raised in a somewhat detective novel style, and after a refreshing and light-hearted mystery-solving style description, Chapter 4 provides clear answers to all the mysteries. This book would be an opportunity for many beginners to start studying martingales in earnest.
As the second theme, a new tool called “stochastic maximal inequality” is introduced in Chapter 5, and it is applied to sparse estimation in Chapter 6. It would be Dr. Nishiyama’s greatest joy if the publication of this book could bring a breath of fresh air to high-dimensional statistics and attract new researchers to this field.
This book assumes that readers have studied the Lebesgue integral theory, but in fact, an overview of the theory is provided in the appendix to welcome readers who had studied it but found it difficult to understand. Some exercises and sample solutions are also provided to check the reader’s understanding of the content.
[Message from Dr. Nishiyama]
Back in the early 1990s, I bought “Markov Chains,” one of the books in the Mathematics One Point series. Fortunately, it was a briskly written short monograph, and it was the only book I, a somewhat lazy student at that time, was able to read through to the end during my time at undergraduate course of university.
Time has passed, and now I have published a new book titled “Martingales” as one of the Statistics One Point series. As a student, I never dreamed that I would be able to write a book in the future, so I am grateful for this series of coincidences and good fortune. I would be happy if this short book is a good “encounter” in some way for those who pick it up.
Nowadays, with the development of the Internet and other things, there are many useful sources for easy research. However, I believe that “books” will continue to play an important role in the future as a universal place of encounter that can be a crossroads in life. I hope that this book will “encounter” many readers.
Click here to purchase the book.
