By Gennady Samorodnitsky

Lengthy variety Dependence is a panoramic survey of the guidelines, types and methods linked to the thought of lengthy reminiscence. it is going to function a useful reference resource for researchers learning lengthy diversity dependence, for these development lengthy reminiscence types, and for those who are attempting to notice the prospective presence of lengthy reminiscence in information.

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4), regardless of the summability of the correlations. Such situations are, clearly, possible. A simple example is ρn = sin na/na, n = 1, 2, . . for 0 < a < π. 7), as another simple example ρn = (−1)n /2, n = 1, 2, . . shows. 7) we need to concentrate on the spectrum of the covariance function of the process. Recall that the spectral measure F is a measure on (−π, π], satisfying Rk = (−π,π] cos(kx) F (dx) for k ≥ 0. 3), then the spectral measure has a continuous density with respect to the Lebesgue measure on (−π, π], the spectral density, given by f (x) = σ2 2π ∞ 1+2 ρn cos nx , −π < x < π .

On the other hand, let k be the Hermite rank of the function g. 15)), as the following example indicates. 4. Take a > 0 such that ∞ ae−a xex e−x a 2 /2 a dx = x2 e−x 2 /2 0 and define g(x) = − a1 x if 0 ≤ x < a . ex−a if x ≥ a dx , 42 Second-Order Theory Set g(x) = −g(−x) for x < 0. Clearly, the function g is odd, measurable, one-to-one, and Eg(X)2 < ∞. Furthermore, by the choice of the number a, E Hn (X)g(X) = 0 for n = 1, and, by the fact that g is odd, the same is true for n = 2. Therefore, the Hermite rank of the function g is at least 3 (in fact, it is not difficult to check that in this case the rank k of g is exactly equal to 3, but one can modify the construction and obtain one-to-one functions of arbitrarily high rank).

5 Second-Order Theory By far the most popular point of view on long range dependence is through a slow decay of correlations. This is related to the original explanation of the Hurst phenomenon by Mandelbrot, discusses in Section 2, and to the simple fact that correlations are one of the easiest to understand and estimate features of a stochastic model. Clearly, such approaches to the notion of long memory are restricted to secondorder stationary processes, and this is the assumption that will made throughout this section.