LONG-MEMORY PROCESSES: PROBABILISTIC PROPERTIES AND STATISTICAL METHODS, by Jan Beran, Yuanhua Feng, Sucharita Ghosh, and Rafal Kulik. Published by Springer London, 2013. Total number of pages: 884. ISBN: 978-3-642-35511-0 (print), 978-3-642-35512-7


In the last 20 years or so, long-memory processes have become a subject of very intensive research in probability
and statistics. The reason could be that the property of long memory is observed in many real time series from
various fields such as geophysics, hydrology, telecommunications, economics, finance and climatology. Many
classical limit theorems no longer are applicable in the case of long-memory stationary random processes. For
instance, the square root laws of central limit theorems are not true anymore when the second-order spectrum
of the stationary processes is not bounded; this causes several analytical difficulties in the derivation of analytic
expressions and proofs associated with estimates obtained in the case of long-memory stationary processes. This
book provides a comprehensive review of these, including a thorough discussion of mathematical and probabilistic
foundations and statistical methods. The main focus is on processes with long memory, self-similar scaling or
fractal properties and power laws. One can say that the book is too long, containing almost 900 pages. The list of
chapters and sections is given subsequently.
Three chapters underlie the core of the book, and they are as follows: limit theorems, statistical inference for
stationary processes and statistical inference for non-stationary processes.
Limit theorems are very important since most statistical procedures in time-series analysis are based on
asymptotic results. Besides basic principles and results, this chapter covers areas where the derivation of
limit theorems for long-memory processes are needed. Although a substantial part of theorems given are for
time-domain-based methods, one can find results for Fourier- and wavelet-transforms based methods as well.
The chapter on statistical inference for stationary processes deals with inference for long-range dependent
(LRD) linear and subordinated processes. Estimation and asymptotic theory of location parameters (mean) and
scale parameters (standard deviation) are considered in this chapter. The estimation of the long-memory parameter
is studied in details including the original R/S method, variance plot, the Kwiatkowski–Phillips–Schmidt–Shin
statistics, the rescaled variance method, detrended fluctuation analysis and temporal aggregation. The refined
estimation procedures like maximum likelihood estimation and its approximate version of the Whittle estimator
are also considered.
The chapter on statistical inference for non-stationary processes discusses the linear, polynomial and nonlinear
regression with LRD errors, and also the authors discussed differences between long-memory, local stationary and
non-stationary processes. The important problem of optimal bandwidth (depending on the long-memory parameter) necessary for smoothing in nonlinear regression is discussed. Some interesting particular cases and examples
are also included.
This book is well written, and the presentation is lucid so that the time-series persons who are interested in
applications can easily follow. The authors have assumed that the readers are familiar with real analysis, probability
theory and statistical inference, and have a good background of time-series methodology. Students doing a PhD
and research workers must be in a position to understand the contents. The historical remarks and presentation
of introductory material to understand proofs must definitely help the readers in understanding proofs and basic
ideas of proofs. Several data sets have been analyzed to illustrate the methodology. This book will be a valuable
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