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Figure 44 Demonstration of the multifractal behavior of a binomial measure b (left) and its distribution function Mb (right) On the top a numerical simulation, ie (452) on the left and Mb (k2 n ) on the right for n = 20 In the middle the moduli of a continuous wavelet transform [DAU 92] where the second Gaussian derivative was taken as the analyzing wavelet (t) for b , resp the third derivative for Mb The dark lines indicate the lines of maxima [JAF 97, BAC 93], ie the locations where the modulus of (2j t s) b (dt) has a local maximum as a function of s with j xed On the bottom a multifractal analysis in three steps First, a plot of log S n w (q) against n tests for linear behavior for various q Second, the partition function (q) is computed as the slopes of a least square linear t of log S n Finally, the Legendre transform (a) of (q) is calculated following (449) Indicated with dashes in the plots of (q) and (a) of b are the corresponding function for Mb , providing empirical evidence for (476), (477), and (483)

Now, setting g(t) = 2n/2 (2n t k) for a smooth analyzing wavelet we have g (t) = 23n/2 (2n t k) and obtain Cn,k ( , ) = 2n/2 (2n k) 2n Cn,k ( , M) (482)

(1 t)2n and assuming exponential decay Estimating 2n kn = 2n t2n of (t) at in nity allows us to conclude w(t) , = 1 + w(t) ,M , and similarly to w(t) COROLLARY 46 f , (a) = f ,M (a + 1) , (a) = ,M (a + 1) (484) (483)

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This is impressively demonstrated in Figure 44 We should note that has one more vanishing moment than which is easily seen by integrating by parts Thus, it

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is natural to analyze the integral of a process, here the distribution function M of the measure , using since the degree of the Taylor polynomials typically grows by 1 under integration NOTE 45 (Visibility of singularities and regularity of the wavelet) It is notable that the Haar wavelet yields the full spectra of the binomial Mb (and also of its distributional derivative b ) This fact is in some discord with the folklore saying that a wavelet cannot detect degrees of regularity larger than its own In other words, a signal will rarely be more regular than the basis elements it is composed of To resolve the apparent paradox, recall the peculiar property of multiplicative measures which is to have constant Taylor polynomials So, it will reveal its scaling structure to any analyzing wavelet with = 0 No higher regularity, ie tk (t)dt = 0 is required The correct reading of the literature is indeed, that wavelets are only guaranteed to detect singularities smaller than their own regularity 47 Self-similarity and LRD The statistical self-similarity as expressed in (41) makes FBM, or rather its increment process, a paradigm of long range dependence (LRD) To be more explicit let denote a xed lag and de ne fractional Gaussian noise (FGN) as G(k) := BH ((k + 1) ) BH (k ) (485)

Possessing the LRD property means that the auto-correlation rG (k) := E [G(n + k)G(n)] decays so slowly that k rG (k) = The presence of such strong dependence bears an important consequence on the aggregated processes G(m) (k) := 1 m

(652)

with a0 = and aK+1 = + , and where ai is an increasing sequence of realities By taking (647), we arrive at the following de nition

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