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Multifractal Scaling: General Theory and Approach by Wavelets
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484 LRD and estimation of warped FBM Let G(k) := B((k + 1)2 n ) B(k2 n ) be FGN in multifractal time (see (493) for the case H = 1/2) Calculating auto-correlations explicitly shows that G is second order stationary under mild conditions with HG = T M (2H) + 1 2 (4100)
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Let us discuss some special cases For example, in a continuous, increasing warp time M, we have always T M (0) = 1 and T M (1) = 0 Exploiting the concave shape of T M we nd that H < H G < 1/2 for 0 < H < 1/2, and 1/2 < H G < H for 1/2 < H < 1 Thus, multifractal warping cannot create LRD and it seems to weaken the dependence as measured through second order statistics Especially in the case of H = 1/2 ( white noise in multifractal time ) G(k) becomes uncorrelated This follows from (4100) Notably, this is a different statement from the observation that the G(k) are independently conditioned on M (see section 481) As a particular consequence, wavelet coef cients will decorrelate fast for the entire process G, not only when conditioning on M (see Figure 45(d)) This is favorable for estimation purposes as it reduces the error variance Of greater importance, however, is the warning that the vanishing correlations should not lead us to assume the independence of G(k) After all, G becomes Gaussian only lead us to assume that we know M A strong, higher order dependence in G is hidden in the dependence of the increments of M which determine the variance of G(k) as in (493) Indeed, Figure 45(c) shows clear phases of monotony of B indicating positive dependence in its increments G, despite vanishing correlations Mandelbrot calls this the blind spot of spectral analysis 49 Bibliography
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Scaling, Fractals and Wavelets
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