Selfsimilar Processes
By addebook • Oct 29th, 2008 • Category: Mathematics •
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Selfsimilar Processes (Princeton Series in Applied Mathematics)
Selfsimilar Processes (Princeton Series in Applied Mathematics)
By Paul Embrechts
Publisher: Princeton University Press
Number Of Pages: 152
Publication Date: 2002-07-16
ISBN-10 / ASIN: 0691096279
ISBN-13 / EAN: 9780691096278
Binding: Hardcover
The modeling of stochastic dependence is fundamental for understanding random systems evolving in time. When measured through linear correlation, many of these systems exhibit a slow correlation decay–a phenomenon often referred to as long-memory or long-range dependence. An example of this is the absolute returns of equity data in finance. Selfsimilar stochastic processes (particularly fractional Brownian motion) have long been postulated as a means to model this behavior, and the concept of selfsimilarity for a stochastic process is now proving to be extraordinarily useful. Selfsimilarity translates into the equality in distribution between the process under a linear time change and the same process properly scaled in space, a simple scaling property that yields a remarkably rich theory with far-flung applications.
After a short historical overview, this book describes the current state of knowledge about selfsimilar processes and their applications. Concepts, definitions and basic properties are emphasized, giving the reader a road map of the realm of selfsimilarity that allows for further exploration. Such topics as noncentral limit theory, long-range dependence, and operator selfsimilarity are covered alongside statistical estimation, simulation, sample path properties, and stochastic differential equations driven by selfsimilar processes. Numerous references point the reader to current applications.
Though the text uses the mathematical language of the theory of stochastic processes, researchers and end-users from such diverse fields as mathematics, physics, biology, telecommunications, finance, econometrics, and environmental science will find it an ideal entry point for studying the already extensive theory and applications of selfsimilarity.
Summary: A beautiful little book…
Rating: 5
This really is a beautiful little book and it ties together well properties which are used and seen in actual data (network traffic, financial returns, etc) with rigorous mathematics.
If you have a project to be completed tomorrow for a customer on estimating network traffic, then don’t start here. Go to one of the more practically oriented books, or to IEEE papers. However, if you are a mathematician, and you are struggling to reform yourself and do something useful with your life (just kidding, all of you research mathematicians out there) and you have a project which involves self similarity, or long range dependence, then this book is ideal for you. It is beautifully written andf ties together all of the practical terms which you have been reading about, but in a very rigorous way. So if you are mathematically inclined, this book is ideal for you, whether or not you have an actual data set to work on.
I don’t have the book by Beran (yet) to know if that book does all of the rigorous mathematics, and also deals with actual data. So perhaps that book does “everything”, but this book is nice for what it does do. Even these authors seem to be struggling hard to get some applied-type material in there, with little pictures of correlations, to illustrate the different decay rates for self-similar processes with different values for the Hurst parameter. Actually, there is a chapter (Chap. 7) on simulation of self-similar processes, and also a chapter (Chap. 
on estimation and estimation of the Hurst exponent. There are plenty of references in these chapters also to guide you to other material.
In summary, “Two thumbs up,” very nicely written, and fills in gaps in more applied literature concerning self-similarity, long range dependence, Hurst parameters. The book will allow you amaze and astound your friends at parties by being able to prove to the network engineers what kinds of processes exhibit self-similarity, with independent and stationary increments (answer is stable Levy processes), or even fundamental facts like why a Hurst parameter exists (Lamperti, 1962 Semi stable processes, Trans Amer. Math Soc.). Believe me, these “tricks” are especially amazing at the pub after a few drinks — I’ve even witnessed the former question about ss-i-si (above) in a Pub Quiz. This book is the “how-to” guide for these amazing feats — get reading!!
Summary: Helps to clarify and organize the subject.
Rating: 5
Selfsimilar processes are stochastic processes that are invariant in distribution under suitable scaling of time and/or space. Fractional Brownican motion is perhaps the best known of these, and it is used in telecommunication and in stochastic integration. Other more recent aplications include finance.
While the underlying idea behind all of this is quite simple and can be traced back to Kolmogorov, it is only recently, with the advent of wavelet methods, that the *computational* power has come into focus.
At the same time, this connection to wavelet analysis is now bringing the *Hilbert space theoretic features* of the subject back to the fore. Amusingly, this was in fact a dominant feature which motivated both A. N. Kolmogorov and Norbert Wiener in the early days; e.g., curves in Hilbert space.
I was pleased with this lovely little book, as it brings out beautifully these two aspects of the subject; and I expect that the book will go over well in the classroom. And the book should help bridge mathematical analysis, probability, and applications.
Reviewed by Palle Jorgensen, September 2004.
Summary: Proofs and properties, nothing else.
Rating: 1
I doubt the authors have ever analyzed data to understand what self-similarity means. In this book, you’ll only find a few proofs and mathematical properties concerning self-similar processes. You won’t find anything else unfortunately. If the aim of the authors was to fill some gap in the scientific literature, i’m wondering which one it is. Better treatments of self-similarity have already been written by Mandelbrot (his recently published books by Springer-Verlag on self-affinity, fractals and multifractals), Sornette (critical phenomena in natural sciences) and Samorodnitsky and Taqqu (stable non-gaussian random processes), and others…
The two most introductory books (and references) on self-similarity and long-range dependence are “Self-Similar Network Traffic and Performance Evaluation” and “Theory and applications of Long-range dependence”. This book is not only too math-oriented but also does not discuss anything related to the physical properties of self-similar processes.
Don’t plan to understand anything to self-similarity by reading this book.
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