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.