Scaling, Fractals and Wavelets

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This last result stems directly from the coupling of the two previous propositions For processes with nite variance (ie, whose third order moment 2 exists) Gaussian processes, just as the FBM, for instance this relation takes on the following speci c form: E |dX (j, k)|2 = E |dX (0, 0)|2 2 j(2H+1) (224)

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Given that the latter are second order statistics, the particular form (24) of the covariance structure of a H ASAS process makes it possible to deduce the asymptotic behavior of the dependence structure of the wavelet coef cients [FLA 92, TEW 92] PROPOSITION 24 The asymptotic covariance structure of the wavelet coef cients of a process X which is zero-mean, self-similar of index H, of nite variance and with stationary increments (H ASAS ) takes on the form: E dX (j, k)dX (j , k ) |2 j k 2 j k |2(H n ) , |2 j k 2 j k |

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which illustrates, on the one hand, that the larger the number of vanishing moments, the shorter the range of the correlation and on the other hand, that if H > n + 1 , 2 the long-range dependence which exists for the increment process if H > 1 , is 2 transformed into a short-range dependence [ABR 95, FLA 92, TEW 92] The set of the results which have just been presented can be made more precise when we specify the distribution law which underlies the self-similar process with stationary increments The Gaussian case, illustrated by the FBM, has been widely studied [FLA 92, MAS 93] Its wavelet coef cients are Gaussian at all scales More recently, interest in the non-Gaussian case has led to developments for self-similar -stable processes (or -stable motions) [ABR 00a, DEL 00] Hence, we can deduce from the wavelet decomposition of such processes that the series of their coef cients dX (j, k), in addition to the above-mentioned properties, is itself -stable with the same index 242 Long-range dependence As speci ed in section 223, stationary processes with long-range dependence are characterized by a slow decrease of their correlation function cX ( ) cr | | , 0 < < 1 Thus, the strong statistical connections maintained even between distant samples, X(t) and X(t + ), make the study and analysis of such processes much more complex, by impairing, for example, the convergence of algorithms relying on empirical moment estimators It will be shown hereafter that wavelet decomposition of a process with long-range dependence makes it possible to circumvent this dif culty since under certain conditions the series of coef cients dX (j, k) exhibit

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Scale Invariance and Wavelets

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short-term dependence The covariance function of the wavelet coef cients possesses the following form: E dX (j, k)dX (j , k ) 2

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j+j 2

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cr | | cf

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(225)

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= 2

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(2 j f ) (2 j f ) i2 f (2 j k 2 j e |f |

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indicating that its asymptotic behavior, ie for the large values of the interval |2 j k 2 j k |, is equivalent to that of its original Fourier transform and hence to that of the relation: 2 (j+j )n 2( j j )n |f |2n | (2 j f ) (2 j f )| = f 0 |f | |f | |f | 2n Thus, we can observe the effect of the number of vanishing moments n of the wavelet, which may compensate the original divergence of the spectrum density of /2, the long-range dependence the process By choosing a wavelet such that n of the process is no longer preserved in the coef cient sequences of the decomposition Hence, the bigger n is, the faster the residual correlation decreases: E dX (j, k)dX (j , k ) |2 j k 2 j k | 2n 1 , |2 j k 2 j k |

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From equation (225) we can also prove that the variance of the wavelet coef cients follows a power law behavior as a function of scales: E |dX (j, k)|2 = 2 j(1 ) cf | (f )|2 df = c0 2 j |f | (226)

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This relation will be at the core of the estimation procedure of the parameter (see the following section) Finally, it is important to specify that, since it is possible to invert the wavelet decompositions (see equation (223)), the non-stationarity of the studied processes does not disappear from the analysis (no more than the long-range dependence does); all the information is preserved but redistributed differently amongst the coef cients Thus, long-range dependence and non-stationarity are related to the approximation coef cients aX (j, k) of the decomposition, whereas self-similarity is observed through the scales, by an algebraic progression of the moments of order q of the detail coef cients dX (j, k)

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