Long range dependence by Gennady Samorodnitsky

By Gennady Samorodnitsky

Lengthy diversity Dependence is a breathtaking survey of the information, versions and strategies 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 versions, and for those who try to become aware of the potential presence of lengthy reminiscence in information.

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We proceed with a construction of a function g1 . Define g1 (x) = 22j if 2−j ≤ x ≤ 2−j + 2−2 j for j = 0, 1, . . 18) holds. Further, j ∞ π 2−j +2−2 2j cos nx g1 (x) dx ≤ 0 2 2 = n ≤ = 2 n 2 n + cos nx dx 2−j j=0 ∞ 22j sin n −2j j 2 cos(n(2−j + 2−2 −1 )) 2 22j sin n −2j 2 2 j=0 ∞ j=0 22j sin j≤log2 log2 n 2 n n −2j 2 2 22j sin j>log2 log2 n n −2j 2 2 . 40 Second-Order Theory Clearly, 2 n 22j sin j≤log2 log2 n n −2j 2 2 ≤ 2 n 22j j≤log2 log2 n ≤ c n−1 log2 n 2 for some 0 < c < ∞ and 2 n 22j sin j>log2 log2 n n −2j 2 2 22j 2 = −2j ≤ 2 n 22j j>log2 log2 n ≤ c n−1 log2 n 2 n −2j 2 2 for some 0 < c < ∞ j>log2 log2 n as well.

This program has the advantage of being applicable to stationary processes with or without finite second moment. The boundary between short and long memory is, further, given by a single number — a certain critical value of the scaling exponent. This last feature is also a drawback of the approach: a single number does not usually represent well the dependence structure of a stochastic process, despite the example of certain Gaussian models. Another drawback of this approach is that a reasonably limited family of the models is thus considered — the increments of self-similar stationary increments processes.

Attractiveness of using the increment processes of self-similar processes with stationary increments as “canonical” models with shorter or longer types of memory is particularly obvious because such processes turn out to be the only possible weak limit in a common class of limiting procedures. Specifically, let (U (t), t ≥ 0) be a stochastic process, and an ↑ ∞ be a sequence of positive numbers.

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