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This means that for all j's such that n1 2 j nA 2 j nB n2 , the wavelet coef cients of X will reproduce the power law: EdX j; k 2 9 2ja cf C a; c0 . Strictly speaking, this last relation holds for wavelets whose frequency support is nite, but it is generally valid to an excellent approximation. 1=f -type processes with a < 1 and n1  0 can be seen as the special case of LRD processes. Note that the de nition of 1=f processes naturally extends to include a < 0.  Let X be such that GX n $ cf jnj a , n 3 0, a ! 0. For a ! 1, the variance does not exist (the integral of the spectrum diverges). X can, however, be seen as a generalized second-order stationary 1=f -type process, in the sense that the variance of the wavelet coef cients remains nite, EdX j; k GX n 2 jC0 2 n j dn 2 cf
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on condition that N > a 1 =2. This is possible as the power-law decrease of the spectrum of the wavelet at the origin jC0 n j % nN , jnj 3 0 balances the divergence of GX n (see Abry et al. [3, 4] for details). Then, just as before, we have EdX j; k 2 $ 2ja cf C a; c0 , j 3 I.  Let X be such that GX n $ cf jnj a , n 3 I, a ! 1, (i.e., n2 I). Its autocovariance function reads EX t X t t $ s2 1 Cjtj2h , t 3 0, with h a 1 =2. Equivalently, it implies that E X t t X t 2 % jtj2h , t 3 0. If X is moreover Gaussian, this implies that the sample path of each realization of the process is fractal, with fractal dimension (strictly speaking Hausdorff dimension) D 5 a =2 [28]. This means that the local regularity of the sample path of the process or, equivalently, its local correlation structure exhibits scaling behavior. Such processes are called fractal.
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2.2 WAVELET AND SCALING: THEORY
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Fractality is reproduced in the wavelet domain (generalization of P1) through EdX j; k 2 % 2j 2h 1 , j 3 I, or equivalently for the CWT: EjTX a; t j2 % a2h 1 , a 3 0 [35, 26], which allows an estimation of the fractal dimension through that of the scaling exponent a 2h 1 5 2D. 2.2.3.5 Summary for Scaling Processes Let X be either an H-sssi p process, or a LRD process, or a (possibly generalized) second-order stationary 1=f -type process or a fractal process. Then the wavelet coef cient, due to the combined effects of F1 and F2, will exhibit the two following properties, which will play a key role in the estimation of the scaling exponent presented below:  P1: The fdX j; k ; k P Zg is a stationary process if N ! a 1 =2 and the variance of the dX j; k accurately reproduces, within a given range of octaves j1 j j2 , the underlying scaling behavior of the data: EdX j; k 2 2ja cf C a; c0 ; where (i) in the case of an H-sssi(p) process, a 2H 1, C a; c0 is to be identi ed from Eq. (2.12), and j1 I and j2 I; (ii) in the case of an LRD process, a is de ned as in Eq. (2.6), C a; c0 is to be identi ed from Eq. (2.16), and j2 I and j1 is to be identi ed from the data; (iii) in the case of a (generalized) second-order stationary 1=f -type process, a is de ned from GX n cf jnj a , n1 jnj n2 , C a; c0 a jnj jC0 n j2 dn, and j1 ; j2 are to be derived from n1 ; n2 ; (iv) in the case of a fractal process, a 2h 1, expressions for C a; c0 can be found in Flandrin and Goncalves [35, 36] and j1 1 and j2 is to be identi ed from the data. 2:18
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 P2: fdX j; k , k P Zg is stationary and no longer exhibits long-range statistical dependences but only short-term residual correlations; that is, it is short-range dependent (SRD) and not LRD, on condition that N ! a=2. Moreover, the higher N the shorter the correlation: EdX j; k dX j; k H % jk k H ja 1 2N ; jk k H j 3 I: Note that these two properties of the wavelet coef cients do not rely on an assumption of Gaussianity. In P2 above, we used only weak reformulations (setting j jH ) of P2 SS and P2 LRD. Their general versions ( j not necessarily equal to jH ) can be used to formulate a stronger idealization of strict decorrelation: ID1: EdX j; k dX jH ; k H 0 if jH ; k H T j; k .
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