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An Introduction to Benford's Law, by Arno Berger, Theodore P. Hill
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This book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law's colorful history, rapidly growing body of empirical evidence, and wide range of applications.
An Introduction to Benford's Law begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford's law.
Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The text includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. This text can serve as both a primary reference and a basis for seminars and courses.
- Sales Rank: #1436493 in Books
- Published on: 2015-05-26
- Original language: English
- Number of items: 1
- Dimensions: 9.30" h x .80" w x 6.40" l, .0 pounds
- Binding: Hardcover
- 256 pages
Review
"This is a marvelous and excellent introduction."--Adhemar Bultheel, European Mathematical Society Bulletin
"A must-read for novices and experts alike. It can be used for a graduate-level topics course or as a reference text for researchers in the field. The exposition is outstanding, with hundreds of carefully chosen examples, figures and diagrams to illustrate the theory. For those who are up for a challenge, the book contains several open problems as well. An Introduction to Benford's Law will surely be the go-to text on the subject for years to come."--Pieter C. Allaart, Mathematical Reviews
From the Back Cover
"This comprehensive book is a gem from an academic research perspective. Researchers in the field need now just look in one place for the mathematical foundations of Benford's law."--Mark J. Nigrini, author of Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection
"This book will become a standard reference on Benford's law. It brings together many results that are currently scattered among several different formats, including some very recent findings. The presentation is accessible, with clear and intuitive statements of results and lucid proofs."--Walter Mebane, University of Michigan
About the Author
Arno Berger is associate professor of mathematics at the University of Alberta. He is the author of Chaos and Chance: An Introduction to Stochastic Aspects of Dynamics. Theodore P. Hill is professor emeritus of mathematics at the Georgia Institute of Technology and research scholar in residence at the California Polytechnic State University.
Most helpful customer reviews
3 of 3 people found the following review helpful.
A monograph on the mathematical theory of Benford's Law
By Frank A. Benford
This book is intended to be a fairly complete compilation of the mathematical theory of Benford's Law. As the authors note (page 223), "the main goal of this book has been to develop a solid and comprehensive theoretical basis for the appearance of Benford's law in a broad range of mathematical settings." So far as I can judge, they have succeeded. Similarly, in the Preface they write "this text is a self-contained comprehensive treatment of the theory of Benford's law that includes formal definitions and proofs, open problems, dozens of basis theorems we discovered in the process of writing that have not before appeared in print, and hundreds of examples."
I've written a fairly lengthy online review of this book for the Mathematical Association of America. The URL for this review is http://www.maa.org/press/maa-reviews/an-introduction-to-benfords-law.
A number of technical comments are found in my review. My only general comment is that the title is somewhat misleading in that there's very little of an introductory nature in this book. The prerequisite to get much out of this book is a fairly deep grounding in measure theory based probability theory. A thorough familiarity with Fourier analysis is required at some points, and ergodic theory is used at others. Hence, the book will be of greatest interest to graduate students in mathematics looking for a dissertation topic, and professional mathematicians looking for a one volume summary of what is now known about the mathematics of Benford's Law.
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