Long-memory processes are ubiquitous in scientific disciplines and applied fields. Long-memory (or long-range dependence) processes are processes where the property of the present will have a significant effect on the properties of the process in the future. After J. Beran’s monograph, Statistics for long memory processes. (1994; Zbl 0869.60045), there are enormous progresses on the probabilistic foundations and statistical principles, new techniques to derive limit theorems, parametric, nonparametric, semiparametric and adaptive inference for stationary, nonstationary, locally stationary and nonlinear processes. This book aims to cover probabilistic and statistical aspects of long-memory processes in as much detail as possible, including a broad range of topics. The authors did an excellent job to reach their goals, and the book would be a must for researchers interested in long-memory processes and practioners on time series and data analysis.
Simon Newcomb, a Canadian-American astronomer and mathematician. worked on the measurement of the position of planets as an aid to navigation, becoming increasingly interested in theories of planetary motions. Newcomb was aware that the table of lunar positions calculated by Peter Andreas Hansen was in error when he visited Paris, France in 1870. Newcomb (1895) realized that errors in astronomy typically effect whole groups of consecutive observations and therefore drastically increase the probable error of estimated astronomical constants (i.e., the ususal \(\sigma /\sqrt{n}\)-rule can no longer be applied). This is the first stationary long-memory processes in the literature. Pearson (1902) carried out experiments simulating astronmical observations to confirm Newcomb’s comments, and showed that observations had their own personal bias and each individual measurement series showed persisting serial correlations. Student (1927) tried to identify sources of errors from (1) the difficulty of obtaining a sample, (2) each operation of the analysis and indicated that this phenomenon is familiar to those who have had astronomical experiences. Student further illustrated another example in a time series on the estimation of nitrogen by the Kjeldahl method which is standard to determine “amino” nitrogens in organic matter. The similar error (Student called it as semi-constant) not only exists but also persist throughout the day, the week and the month, and it is hard to remove the error even with careful statistical examination. Student’s semi-constant error may be close to slowly decaying autocorrelations. It would be surprised to find any laboratory without these errors. Smith (1938) later formulated this as an empirical law.
One of the most cited examples in long-range dependence is the long-memroy process of the occurrence of flooding of the Nile River. Hurst (1951) in hydrology found an empirical law while investigating the long-term storage capacity of reservoirs for the Nile. He recognized the non-stationarity in his original geophysical series and designed an experioemnt with “probability cards” to produce sudden changes in mean of the process and obtained empirical estimations of the Hurst exponent near 0.71. The Hurst exponent is referred to the index of long-range dependence which quantifies the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction. A Hurst exponent H in the range \(0.5 < H < 1\) indicates a time series with long-term positive autocorrelation, and a Hurst exponent H in the range \(0 < H < 0.5\) indicates a time series with long-term switching between high and low values in adjacent pairs, meaning that a single high value will probably be followed by a low value and that the value after that will tend to be high, with this tendency to switch between high and low values lasting a long time into the future. A Hurst exponent \(H=0.5\) indicates a completely uncorrelated series where the autocorrelations at small time lags can be positive or negative but where the absolute values of the autocorrelations decay exponentially quickly to zero. This in contrast to the typically power law decay for the \(0.5 < H < 1\) and \(0 < H < 0.5\) cases.
Mandelbrot (1982) declared that his original investigation on the Hurst effect was one of sources of inspiration for his fractals or self-similarities. Mandelbrot created the first-ever “theory of roughness”, and emphasized the use of fractals as realistic and useful models for describing many “rough” phenomena in the real world. Mandelbrot and Ness (1968) believed that fractional Brownian motions provide useful models for a host of natural time series and wished to present their curious properties to science with practical interest with a minimum of mathematical difficulty. The basic property of fractional Brownian motions is that the span of interdependence between increments can be infinite. Empirical studies of random chance phenomena often suggest a strong interdependence between distant samples, as in astronomy, chemistry, economy, engineering, ecosciences, hydrology, mathematics, physics and statistics. Granger (1966) discovered the long memory phenomenon in economics and asserted that most economic time series measured by their levels have spectra that exhibit a smooth declining shape with considerable power at very low frequencies. The Granger law is that the long-term fluctuations in economic variables, if decomposed into frequency components, are such that the amplitude of the components decrease smoothly with decreasing periods. Granger and Joyeux (1980) used fractionbal ARIMA models that greatly improved the applicatability of long-range dependence in statistical practice.
The real practical examples of long-memory processes can be continued. The theoretical studies of long-memory processes can be traced back to 17th century. Leibnitz (1646–1716) studied recursive self-similarity, Bolzano (1781–1848) discovered the first fractal, Hausdorff (1918) defined the fractal dimension, and Kolmogorov (1940) studied the mathematical models for long-memory processes in the context of turbulence. Rosenblatt (1961) was among the first ones to derive a noncentral limit theorem where the limiting process is non-Gaussian because of the nonsummable correlations and nonlinearity.
The first two chapters of the book are devoted to the basic definition of long memory, data examples with typical long-memory behaviour, second-order definitions for nonstationary processes, different dependence measures and extended memories. Chapter 2 also provides typical methods for constructing long-memory processes, and nonlinear processes in financial applications. Long memory, short memory and antipersistence are characterized by the spectral density function slowly varying at infinity in Zygmund’s sense. Fractional ARIMA models, other fractionally differenced processes, FEXP processes and fractional Gaussian noise are listed as examples for study. Those volatility models play an important role in understanding the models of assets and pricings of stocks and their derivatives as well as the credit risk in the reduced form. Those long memory processes occurred from particle systems, turbulence, ecological systems, and network traffic models are mentioned in this chapter with enough background literature.
Chapter 3 gives a general description of univariate orthogonal polynomials and multivariate Hermite polynomials, and the notion of Wick products in terms of Appell polynomials. Basic properties of wavelets, the connections and differences between the fractal behavior and the long-range dependence behavior are given in this Chapter. The authors present integral representations of fractional Brownian motion and Hermite-Rosenblatt processes from the Mandlebrot and Ness (1968) paper, briefly review stable random variables, stable Lévy processes and stable random measures to define linear fractional stable motions and fractional calculus.
Chapter 4 deals with asymptotic results in time series analysis. Billingsley’s criterium for tightness, a fundamental central limit theorem for inverse and the specral representations of stationary sequences are presented. For partial sums with finite moments and the asymptotic behavior of variance of the finite sums in long-memory, short-memory and antipersistence cases, the authors present their limiting distributions to Hermite-Rosenblatt processes from Appell’s polynomial approach to a martingale approach. Limiting theorems for sums with infinite moments are also presented, and limiting results for infinite-variance stochastic volatility models with long memory are discussed in a very nice way with a good summary. This may be very helpful for financial engineering to understand the forecasting and market efficiency. The sections on applications and extensions as well for counting processes and traffic models are quite useful.
Chapter 5 is devoted to statistical inference for long-range dependent linear and subordinated processes. Location estimation, tests and confidence intervals based on the sample mean, M-estimations and scale estimations are given in this chapter in detail. The sample variance is not the best choice under long-range dependence and the limiting distribution is quite complicated because it is of the Hermite-Rosenlatt type. Some heuristic and graphic methods commonly used for long-memory identification, including the original method proposed by Hurst and later studied by Mandelbrot, the KPSS statistic, the rescaled variance method, detrended fluctuation analysis and temporal aggregation are explained. From the computational point of view, the Whittle estimator and the AR-estimator are considerably faster than the exact MLE. This is very important in a long-memory setting since the sample mean has a slow rate of convergence. The fast Fourier transform can be used to make computations very fast. Akaike’s information criterion, semiparametric estimation and the fractional autogressive modelling with a growing AR-order and broadband estimation based on FEXP-models are presented.
Chapter 6 considers nonlinear processes with long memory on stochastic volatility models. For stochastic volatility models, direct maximum likelihood estimation is not always feasible due to the presence of an unobserved latent process; for the LARCH models, a maximum likelihood approach is feasible in principle due to the volatility of explicitly past observations. The tail index estimation for heavy-tailed stochastic volatility models is discussed through asymptotic normality of the Hill estimator and numerical estimation. This chapter is relatively shorter and lacks of financial and economic examples and applications for the empirical analysis of financial data.
Chapter 7 deals with statistical inference for nonstationary processes. For statistical inference, including estimation, testing and forecasting, the distinction between stationary and nonstationary and the distinction between stochastic and deterministic components are essential. Standard techniques in nonparametric regression are kernel and local polynomial smoothing with the choice of a suitable bandwidth. It is a formidable task for fractional processes with unknown long-memory parameters. The optimal bandwidth depends on the unknown long-memory parameter, and it is plausible to set data driven algorithms for asymptotically optimal bandwidth selection and simultaneous estimation of dependence parameters. Extension to nonlinear processes with trends are briefly considered, as well as an alternative approach to kernel and local polynomial smoothing, trend estimation based on wavelets and the issue of optimal selection of the number of resolution levels. The CUSUM statistics and the asymptotic results for change points in the mean, changes in the distribution using empirical processes, changes in the spectrum and rapid change points in the trend function are considered in this chapter.
The last three chapters are short and brief on forecasting, spatial and space-time processes, and resampling or bootstrap methods. Prediction for nonlinear processes can differ quite substantially from the case of linear processes. Some basic problems are stated. Either because of authors’ short of energy or the page-limit for the book, the last three chapters present only basic definitions and basic results.
Overall, the book is an excellent choice for anyone who is working in fields related to long-memory processes with many update information and research topics. Clearly the book can be used as a primary source for researchers in the area of long memory processes and related topics, but not a good choice as a textbook for a graduate course on long memory processes. The book contains many examples and theorems, but no exercises and no self-contained proofs for some theorems stated in the book.