Barbu and Limnios’ goal is to present a complete picture of the basic theory of finite state space semi-Markov processes in discrete time, describe its applications to reliability and DNA analysis, and obtain estimation results for hidden semi-Markov models (HSSM).

To cover these goals, the book is divided into six chapters. Chapter 1 presents the main features discussed in the book. Chapters 2 and 3 introduce the basic results of renewal theory in discrete time and the Markov-renewal chain, respectively. Chapter 4 constructs nonparametric estimators for a discrete-time semi-Markov system and their asymptotic properties.

Chapter 5 obtains explicit expressions for the reliability function of discrete-time semi-Markov reliability systems with suitable illustrations. Chapter 6 investigates the asymptotic properties, consistency, and the asymptotic normality of maximum-likelihood estimation (MLE) of HSMM. It also proposes an expectation-maximization (EM) algorithm that allows one to practically find the MLE of HSMM, and applies this to a classical problem in DNA analysis and to CpG island detection. Each chapter contains several exercises. There are five appendices that render the book self-contained.

With this book, the authors make a strong case for the versatility and usefulness of hidden Markov and semi-Markov models in discrete time. The prerequisites are finite space Markov chains and martingales.

I highly recommend this book for applied probabilists and statisticians interested in reliability and DNA analysis, and for theoretically oriented reliability and bioinformatics engineers.