Let {B H (u)} u∈‫ޒ‬ be a fractional Brownian motion (fBm) with index H ∈ (0, 1) and Sp(B H ) be the closure in L 2 ( ) of the span Sp(B H ) of the increments of fBm B H . It is well-known that, when B H = B 1/2 is the usual Brownian motion (Bm), an element X ∈ Sp(B 1/2 ) can be characterized by a unique function f X ∈ L 2 ‫,)ޒ(‬ in which case one writes X in an integral form as X = ‫ޒ‬ f X (u)dB 1/2 (u). From a different, though equivalent, perspective, the space L 2 ‫)ޒ(‬ forms a class of integrands for the integral on the real line with respect to Bm B 1/2 . In this work we explore whether a similar characterization of elements of Sp(B H ) can be obtained when H ∈ (0, 1/2) or H ∈ (1/2, 1). Since it is natural to define the integral of an elementary function f = n k=1 f k 1 [u k ,u k+1 ) by n k=1 f k (B H (u k+1 ) − B H (u k )), we want the spaces of integrands to contain elementary functions. These classes of integrands are inner product spaces. If the space of integrands is not complete, then it characterizes only a strict subset of Sp(B H ). When 0 < H < 1/2, by using the moving average representation of fBm B H , we construct a complete space of integrands. When 1/2 < H < 1, however, an analogous construction leads to a space of integrands which is not complete. When 0 < H < 1/2 or 1/2 < H < 1, we also consider a number of other spaces of integrands. While smaller and hence incomplete, they form a natural choice and are convenient to work with. We compare these spaces of integrands to the reproducing kernel Hilbert space of fBm.

This modern and comprehensive guide to long-range dependence and self-similarity starts with rigorous coverage of the basics, then moves on to cover more specialized, up-to-date topics central to current research. These topics concern, but are not limited to, physical models that give rise to long-range dependence and self-similarity; central and non-central limit theorems for long-range dependent series, and the limiting Hermite processes; fractional Brownian motion and its stochastic calculus; several celebrated decompositions of fractional Brownian motion; multidimensional models for long-range dependence and self-similarity; and maximum likelihood estimation methods for long-range dependent time series. Designed for graduate students and researchers, each chapter of the book is supplemented by numerous exercises, some designed to test the reader's understanding, while others invite the reader to consider some of the open research problems in the field today.

Operator fractional Brownian motions (OFBMs) are (i) Gaussian, (ii) operator self-similar and (iii) stationary increment processes. They are the natural multivariate generalizations of the well-studied fractional Brownian motions. Because of the possible lack of time-reversibility, the defining properties (i)-(iii) do not, in general, characterize the covariance structure of OFBMs. To circumvent this problem, the class of OFBMs is characterized here by means of their integral representations in the spectral and time domains. For the spectral domain representations, this involves showing how the operator self-similarity shapes the spectral density in the general representation of stationary increment processes. The time domain representations are derived by using primary matrix functions and taking the Fourier transforms of the deterministic spectral domain kernels. Necessary and sufficient conditions for OFBMs to be time-reversible are established in terms of their spectral and time domain representations. It is also shown that the spectral density of the stationary increments of an OFBM has a rigid structure, here called the dichotomy principle. The notion of operator Brownian motions is also explored.

Let B H be a fractional Brownian motion with self-similarity parameter H P (0, 1) and a . 0 be a ®xed real number. Consider the integral a 0 f (u)dB H (u), where f belongs to a class of non-random integrands Ë H,a . The integral will then be de®ned in the L 2 (Ù) sense. One would like Ë H,a to be a complete inner-product space. This corresponds to a desirable situation because then there is an isometry between Ë H,a and the closure of the span generated by B H (u), 0 < u < a. We show in this work that, when H P ( 1 2 , 1), the classes of integrands Ë H,a one usually considers are not complete inner-product spaces even though they are often assumed in the literature to be complete. Thus, they are isometric not to spfB H (u), 0 < u < ag but only to a proper subspace. Consequently, there are (random) elements in that closure which cannot be represented by functions f in Ë H,a . We also show, in contrast to the case H P ( 1 2 , 1) that there is a class of integrands for fractional Brownian motion B H with H P (0, 1 2 ) on an interval [0, a] which is a complete inner-product space.

Let α ∈ (1, 2) and X α be a symmetric α-stable (SαS) process with stationary increments given by the mixed moving averagewhere (X, X , µ) is a standard Lebesgue space, G : X × R → R is some measurable function and M α is a SαS random measure on X ×R with the control measure m α (dx, du) = µ(dx)du. We show that if X α is self-similar, then it is determined by a nonsingular flow, a related cocycle and a semi-additive functional. By using the Hopf decomposition of the flow into its dissipative and conservative components, we establish a unique decomposition in distribution of X α into two independent processeswhere the process X D α is determined by a nonsingular dissipative flow and the process X C α is determined by a nonsingular conservative flow. In this decomposition, the linear fractional stable motion, for example, is determined by a conservative flow.