A number of new books on long-range dependence and self-similarity have been published in the last ten years or so. On the statistics and modeling side, these include (in alphabetical order):

1. • Beran et al. [127], Cohen and Istas [253], Giraitis et al. [406], Leonenko [612], Palma [789].

On the probability side, these include:

1. • Berzin, Latour, and León [138], Biagni et al. [146], Embrechts and Maejima [350], Major [673], Mishura [722], Nourdin [760], Nourdin and Peccati [763], Nualart [769], Peccati and Taqqu [797], Rao [842], Samorodnitsky [879, 880], Tudor [963].

On the application side, these include:

1. • Dmowska and Saltzman [319] regarding applications in geophysics, Park and Willinger [794] and Sheluhin et al. [903] in connection to telecommunications, Robinson [854] and Teyssière and Kirman [958] in connection to economics and finance.

Collections of articles include:

1. • Doukhan, Oppenheim, and Taqqu [328], Rangarajan and Ding [841].

See also the following books of related interest:

1. • Houdré and Pérez-Abreu [488], Janson [529], Marinucci and Peccati [689], Meerschaert and Sikorskii [712], Terdik [954].

We have not covered in this monograph a number of other interesting topics related to long-range dependence and/or self-similarity, including:

1. • Long-range dependence for point processes: In a number of applications, data represent events recorded in time (e.g., spike trains in brain science, packet arrivals in computer networks, and so on). Such data can be modeled through point processes (e.g., Daley and Vere-Jones [282, 283]). Long-range dependence for point processes was defined and studied in Daley [281], Daley and Vesilo [284], Daley, Rolski, and Vesilo [285]. See also the monograph of Lowen and Teich [648].

2. • Other nonlinear time series with long-range dependence: A large portion of this monograph (in particular, Chapter 2) focuses on linear time series exhibiting long-range dependence, with the exception of the non-linear time series defined as functions of linear series in Chapter 5. A number of other nonlinear time series with long-range dependence features have been considered in the literature, especially in the context of financial time series, e.g., FIGARCH and related models (Ding, Granger, and Engle [316], Bail-lie, Bollerslev, and Mikkelsen [89], Andersen and Bollerslev [29], Andersen, Bollerslev, Diebold, and Ebens [30], Comte and Renault [257], Tayefi and Ramanathan [953], Deo et al. [302]). See also Shao and Wu [902], Wu and Shao [1013], Baillie and Kapetanios [87].

In the context of financial time series, long-range dependence has been associated to their volatility process.

There are fundamental connections between self-similarity and fractional calculus, which is an area of real analysis. These connections are explored here in the context of fractional Brownian motion (FBM).

The basics of fractional calculus can be found in Section 6.1 where we introduce the following fractional integrals and derivatives:

1. • Riemann–Liouville fractional integral on an interval (Definition 6.1.1)

2. • Riemann–Liouville fractional derivative on an interval (Definition 6.1.4)

3. • Fractional integral on the real line (Definition 6.1.13)

4. • Liouville fractional derivative on the real line (Definition 6.1.14)

5. • Marchaud fractional derivative on the real line (Definition 6.1.18)

Representations of FBM are obtained in terms of fractional integrals and derivatives in Section 6.2. These representations then play a central role in developing deterministic integration with respect to FBM, and in being able to address various applications involving that process.

Integrals of deterministic functions with respect to FBM will be called fractional Wiener integrals. They are introduced in Section 6.3 and defined first as integrals on an interval (Section 6.3.2) and then as integrals on the real line (Section 6.3.6). Applications involving the Girsanov formula, prediction and filtering can be found in Section 6.4. Integrals of random functions with respect to FBM are introduced in Chapter 7.

Fractional Integrals and Derivatives

We provide here the basics of fractional integrals and derivatives on an interval (Sections 6.1.1 and 6.1.2) and fractional integrals and derivatives on the real line (Sections 6.1.3–6.1.5).

Fractional Integrals on an Interval

The standard way of motivating the definition of a fractional integral is to start with an n–tuple iterated integral and show that it can be expressed as a single integral involving the parameter n. The fractional integral of order α > 0 is then defined by replacing the integer n by the real number α in the resulting expression.

Let a < b be two real numbers and φ be an integrable function on [a, b]. A multiple integral of φ can be expressed as

In this chapter, we present a number of physical models for long-range dependence and self-similarity. What do we mean by “physical models” for long-range dependence? First, models motivated by a real-life (physical) application. Second, though these models lead to long-range dependence, they are formulated by using other principles (than long-range dependence). One of these principles is often “responsible” for the emergence of long-range dependence. For example, the infinite source Poisson model discussed in Section 3.3 below, may model an aggregate traffic of data packets on a given link in the Internet. For this particular model, long-range dependence arises from the assumed heavy tails of workload distribution for individual arrivals. We consider the following models:

1. • Aggregation of short-range dependent series (Section 3.1)

2. • Mixture of correlated random walks (Section 3.2)

3. • Infinite source Poisson model with heavy tails (Section 3.3)

4. • Power-law shot noise model (Section 3.4)

5. • Hierarchical model (Section 3.5)

6. • Regime switching (Section 3.6)

7. • Elastic collision of particles (Section 3.7)

8. • Motion of a tagged particle in a simple symmetric exclusion model (Section 3.8)

9. • Power-law Pólya's urn (Section 3.9)

10. • Random walk in random scenery (Section 3.10)

11. • Two-dimensional Ising model (Section 3.11)

12. • Stochastic heat equation (Section 3.12)

13. • The Weierstrass function connection (Section 3.13)

Having physical models for long-range dependence is appealing and useful for a number of reasons. For example, in such applications, the use of long-range dependent models is then justified and thus more commonly accepted. In such applications, the parameters of long-range dependence models may carry a physical meaning. For example, in the infinite source Poisson model, the long-range dependence parameter is expressed through the exponent of a heavy-tailed distribution.

Aggregation of Short-Range Dependent Series

One way long-range dependence arises is through aggregation of short-range dependent (SRD) series. We focus here on aggregation of AR(1) series only, which is one of the simplest examples of SRD series and which has attracted most attention in the literature on aggregation.

Let a be a random variable supported on (−1, 1) and having a distribution function Fa. Suppose that a(j), j ≥ 1, are i.i.d. copies of a.

We focus in this book on long-range dependence and self-similarity. The notion of long-range dependence is associated with time series whose autocovariance function decays slowly like a power function as the lag between two observations increases. Such time series emerged more than half a century ago. They have been studied extensively and have been applied in numerous fields, including hydrology, economics and finance, computer science and elsewhere. What makes them unique is that they stand in sharp contrast to Markovian-like or short-range dependent time series, in that, for example, they often call for special techniques of analysis, they involve different normalizations and they yield new limiting objects.

Long-range dependent time series are closely related to self-similar processes, which by definition are statistically alike at different time scales. Self-similar processes arise as large scale limits of long-range dependent time series, and vice versa; they can give rise to long-range dependent time series through their increments. The celebrated Brownian motion is an example of a self-similar process, but it is commonly associated with independence and, more generally, with short-range dependence. The most studied and well-known self-similar process associated with long-range dependence is fractional Brownian motion, though many other self-similar processes will also be presented in this book. Self-similar processes have become one of the central objects of study in probability theory, and are often of interest in their own right.

This volume is a modern and rigorous introduction to the subjects of long-range dependence and self-similarity, together with a number of more specialized up-to-date topics at the center of this research area. Our goal has been to write a very readable text which will be useful to graduate students as well as to researchers in Probability, Statistics, Physics and other fields. Proofs are presented in detail. A precise reference to the literature is given in cases where a proof is omitted. Chapter 2 is fundamental. It develops the basics of long-range dependence and self-similarity and should be read by everyone, as it allows the reader to gain quickly a basic familiarity with the main themes of the research area. We assume that the reader has a background in basic time series analysis (e.g., at the level of Brockwell and Davis [186]) and stochastic processes. The reader without this background may want to start with Chapter 1, which provides a brief and elementary introduction to time series analysis and stochastic processes.