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q q S n w, b (q) = 2nq E [|C0,0 |q ] (E [M0 ] + E [M1 ]) , n
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and similar for Mb Provided E [|C0,0 |q ] is nite this immediately gives T w, b (q) + q = T w,Mb (q) = T ,Mb (q), T
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Imposing additional assumptions on the distributions of the multipliers we may n also control wk ( b ) themselves and not only their moments To this end, we should be able to guarantee that the wavelet coef cients do not decay too fast (compare (410)), ie the random factor (462) which appears in (461) does not become too small Indeed, it is suf cient to assume that there is some > 0 such that |C0,0 ( b )| (n,k ) almost surely Then for all t, (1/n) log( (t) b n (dt)) 0, and with (461) 1 w b (t) = lim inf log2 2n/2 |Cn,kn | = Mb (t) 1, n n (478)
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and similarly w b (t) = Mb (t) 1 Observe that this is precisely the relation we expect between the scaling exponents of a process and its (distributional) derivative at least in nice cases and that it is in agreement with (477) In summary ( rst observed for deterministic binomials in [BAC 93]): COROLLARY 45 Assume that b is a random binomial measure satisfying i)-iii) (n,k) (n,k) Assume, that the random variables | (t) b (dt)| resp | (t)Mb (t)dt| are uniformly bounded away from 0 Then, the multifractal formalism holds for the wavelet based spectra of b , resp Mb , ie
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w, dim(Ea b ) = f w, b (a) = w, b (a) = T w, b (a), as as as
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w,M dim(Ea b ) = f w,Mb (a) = w,Mb (a) = T w,Mb (a) as as as (n,k) (n,k)
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Requiring that | (t) b (dt)| resp | (t)Mb (t)dt| should be bounded away from zero in order to insure (478), though satis ed in some simple cases, seems unrealistically restrictive to be of practical use A few comments are in order here First, this condition can be weakened to arbitrarily allow small values of these integrals, as long as all their negative moments exist This can be shown by an argument using the Borel-Cantelli lemma Second, the condition may simplify in two ways For iid multipliers we know that these integrals are equal in distribution to C0,0 , thus only n = k = 0 has to be checked Further, for the Haar wavelet and symmetric multipliers, it becomes simply the condition that M0 be uniformly bounded away from zero (see (460)), or at least that E[|M0 1/2|q ] < for all negative q Third, if we drop iii) and allow the multiplier distributions to depend on scale (see (n,k) (n,k) [RIE 99]), then | (t) b (dt)| resp | (t)Mb (t)dt| has to be bounded away from zero only for large n In applications such as network traf c modeling we nd n+1 n+1 that on ne scales M2k M2k+1 is best modeled by discrete distributions on [0, 1] with large variance, ie without mass around 1/2
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Fourth, another way out is to avoid small wavelet coef cients entirely in a multifractal analysis More precisely, we would follow [BAC 93, JAF 97] and n replace Cn,kn in the de nition of wkn (412) by the maximum over certain wavelet coef cients close to t Of course, the multifractal formalism of section 44 still holds [JAF 97] gives conditions under which the spectrum w, b (a) based on this n modi ed wk agrees with the H lder spectrum dim(Ea ) based on hn (Mb ) k 462 Multifractal properties of the derivative Corollary 45 establishes for the binomial what intuition suggests in general, ie that the multifractal spectra of processes and their derivative should be related in a simple fashion at least for certain classes of processes As we will show, increasing processes have this property, at least for the wavelet based multifractal spectra
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h However, the order of H lder regularity in the sense of the spaces Ct (see Lemma 41) might decrease under differentiation by an amount different from 1 This is particularly true in the presence of highly oscillatory behavior such as chirps , h as the example ta sin(1/t2 ) demonstrates In order to assess the proper space Ct a 2-microlocalization has to be employed For good surveys see [JAF 95, JAF 91]
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In order to establish a general result on derivatives we place ourselves in the framework whereby we care less for a representation of a process in terms of wavelet coef cients and are interested purely in an analysis of oscillatory behavior A typical example of an analyzing mother wavelet are the derivatives of the Gaussian kernel exp ( t2 /2) which were used to produce Figure 44 The idea is to use integration by parts For a continuous measure on [0, 1] with distribution function M(t) = ([0, t)) and a continuously differentiable function g this reads as
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