Distribution theory of runs and patterns and its applications. A finite Markov chain imbedding approach. (English) Zbl 1030.60063

Fu, James C.; Lou, W. Y. Wendy

Singapore: World Scientific (ISBN 981-02-4587-4/pbk; 978-981-277-920-5/ebook). x, 162 p. (2003).

The occurrence of runs and patterns in a sequence of discrete trial outcomes or random permutations is an important concept in various areas of science, including reliability engineering, quality control, psychology and DNA sequence matching. This book is not a review book for the theory of runs and pattern, nor is it intended to be used primarily as a course textbook; it is mainly aimed at researchers in applied statistics and probability who are interested in using the finite Markov chain imbedding technique to study the distributions of runs and patterns arising in specific applications.
This book is organized in the following way. In Chapter 2 the authors introduce the basic ideas and techniques of finite Markov chain imbedding. This chapter lays the foundation for computing the pdfs of runs and patterns including waiting-time distributions. Chapter 3 examines the distributions of runs and patterns associated with two-state trials, and in Chapter 4 the extension to multi-state trials via the forward and backward principle is treated. Chapter 5 mainly studies the waiting-time distributions of simple and compound patterns, as well as their generating functions and large deviation approximations. In Chapter 6, the finite Markov chain imbedding technique is extended to the study of distributions of patterns in random permutations of integers, focusing in detail on the Eulerian and the Simon Newcomb numbers. Chapter 7 covers several applications of the distribution theory of runs and patterns in the areas of the reliability of engineering systems, hypothesis testing, continuity measurement in health care, and quality control.

Reviewer: Nicko G.Gamkrelidze (Moskva)

MSC:

60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
60-02	Research exposition (monographs, survey articles) pertaining to probability theory

Keywords:

theory of runs and patterns; finite Markov chain imbedding