UK Nonlinear News, May 2003
Princeton Series in Applied Mathematics, 2002, 111p.
Hardback, £19.95 (Amazon)
Across the disciplines there is a growing awareness of phenomena for which no characteristic scale is readily identified. Mandelbrot pointed out that spatial fractals are everywhere we look. Temporal fractals are ubiquitous in the form of signals exhibiting 1/f fluctuations. Accordingly a great number of books intended for a general reader, often containing rather imprecise discussion, have been published over the last few years. This concise and succinct book is an attempt to treat selfsimilar processes with precision and rigour. It is a mathematics book written by mathematicians for mathematicians. The exposition follows the standard format expected and loved by mathematicians: enumerated Definitions, Theorems, Lemmas, Remarks, Examples, etc.
The book contains nine chapters. The precise mathematical definition of a selfsimilar process is given in the first chapter, where relations are made to Fractional Brownian Motion (FBM) and Stable Lˆvy Processes. The brief second chapter is called "Some Historical Background" and includes results by Lamperti from the 1960ties on the relation between selfsimilar processes and limit theorems. Also explained is Sinai's result on the uniqueness of fractional Gaussian noise as a fix point for the renormalisation group. Two Chapters are devoted to the discussion of processes which have either independent increments (chapter five) or stationary increments (chapter three). (L\210vy processes are the only selfsimilar processes with independent as well as stationary increments). Chapter three estimates moments and relates them to the exponent of the selfsimilar process. Finite and infinite variance processes are discussed and the role of long-range dependence between increments is dealt with as are the relevant Limit Theorems. Chapter five starts with Sato's selfdecomposition theorem which is followed by theorems concerned with the conditions under which joint distributions are selfdecomposable. The Fractional Brownian Motion process is revisited in detail in chapter four where the following topics are discussed: the sample path, the non-semimartingale property, stochastic integration together with a list of results on the distribution of the maximum of a FBM, occupation times, multiple points and large increments.
Sample path properties for general selfsimilar stable processes with stationary increments are discussed in chapter six. A brief review of how selfsimilar processes are best simulated is given in chapter seven. This chapter covers a great deal of ground in a small amount of space by including extensive references to easily accessible material, including on the web. Chapter eight is the most applied chapter in the book. This chapter includes a brief, but nevertheless worthwhile, description of how long-range correlations in data can be detected. Methods mentioned include Hurt's so called R/S-statistics, or "rescaled adjusted" method, the correlogram, least square regression and maximum likelihood methods. The chapter closes with a brief review of the literature. The final chapter discusses how the concept of selfsimilarity in d-dimensional real space, R, can be extended to allow for scaling of linear operators on R. Further, selfsimilarity is generalised to the concept semi-selfsimilarity which is relevant to diffusion on fractals such as the Sierpinski gasket.
The authors assume detailed familiarity with basic formal mathematical notation and with statistics nomenclature and concepts. The books jacket suggests that the book may be the "ideal entry point for studying the already extensive theory and applications of self-similarity" for researchers and end-users from physics, biology, telecommunication, etc. I doubt this will be the case. Basic notation and concepts are not introduced but assumed known to the reader. There are no attempts to explain, using words or simple examples, either the concepts used or the importance of the results derived. References to applications would have made the book significantly more accessible to non-mathematicians. The authors suggest in their preface that the book should be considered as "intermediate lecture notes trying to bridge the gap between the various existing developments on the subject. The scientific community still awaits a definitive text. We hope that our contribution will be helpful for someone undertaking such an endeavour". I agree entirely. The book is ideal for an undergraduate mathematics course in selfsimilar processes. If someone were to write a book for the broader community then Embrechts and Maejima's book would certainly serve as an excellent review and introduction to the mathematics literature.