2 2H | | = 2

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It is now possible to study in depth its asymptotic behaviors in both limits of large and small s For instance, we show that in the limit + (ie, ), the autocorrelation function decreases asymptotically as 2(H 1) : E X ( ) (t)X ( ) (t + ) 2 2H (2H 1) 2 2(H 1) 2 (26)

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By Fourier duality, this behavior induces an algebraic spectral divergence with exponent 1 2H at the origin Self-similar processes with stationary increments are hence closely related to long-range dependent processes In the other limit, 0 (ie, ), we show that for H > 1 : 2 E X ( ) (t)X ( ) (t + ) 2 2H 1 2H | |2H (27)

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This behavior characterizes the local regularity of each sample path of the process X The following sections explain the notions associated with each of these limits: long-range dependence on the one hand and local regularity on the other hand 223 Long-range dependence DEFINITION 23 A second order stationary process X = {X(t), t } is said to be long-range dependent (or to have long memory ) if its correlation function cX ( ) := E X(t)X(t + ) is such that [BER 94, SAM 94]: cX ( ) cr | | , + (28)

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Scaling, Fractals and Wavelets

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with 0 < < 1 In the same way, the power spectrum density:

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of a long-range dependent process is such that: X (f ) cf |f | , f 0 (29)

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with 0 < = 1 < 1 and cf = 2 (2 ) sin((1 ) /2) ( ) cr , where denotes the usual Gamma function Under its form (28), long-range dependence is related to the fact that, for large lags, the (algebraic) decrease of the correlation function is so slow that it does not enable its summability: hence, there is a long memory effect, in the sense that signi cant statistical relations are maintained between very distant samples Obviously, this situation is in contrast with that of Markovian processes with short memory, which are characterized by an asymptotic exponential reduction of the correlations By de nition, the existence of an exponential decrease involves a characteristic time scale, whereas this is no longer the case for an algebraic decrease: hence, it is a matter of scaling law behavior By Fourier duality, long-range dependence implies that X (0) = , in accordance with the power law divergence expressed by (29) Finally, even if the property of long-range dependence exists and although its de nition is independent from that of self-similarity, relation (26) demonstrates that a strong bond exists between these two notions, since it indicates that the increment process of a self-similar process with stationary increments presents, if H > 1 , long-range dependence 2 224 Local regularity The main issue of this section, rather than the long-term behavior of the autocorrelation function, is its short-term behavior Let X be a second order stationary random process, whose autocorrelation function is originally such that: E X(t)X(t + ) 2 (1 C| |2h ), 0, 0 < h < 1 (210)

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Hence, it is easy to prove that this original covariance structure is equivalent to an algebraic behavior of the increments variance in the limit of short increments: E |X(t + ) X(t)|2 C| |2h , 0, 0 < h < 1

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This relation provides information on the local regularity of each sample path of the process X For Gaussian processes, for instance, it indicates that these sample paths are continuous of order h < h When 0 < h < 1, this means that these trajectories of X are everywhere continuous but nowhere differentiable

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