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[MON 00] M ONTSENY G, AUDOUNET J, M ATIGNON D, Diffusive representation for pseudo-differentially damped non-linear systems , in I SIDORI A, L AMNABHI -L AGARRIGUE F, R ESPONDEK W (Eds), Nonlinear Control in the Year 2000 (Paris, France), Springer-Verlag, vol 2, p 163 182, 2000 [OLD 74] O LDHAM KB, S PANIER J, The Fractional Calculus, Academic Press, New York and London, 1974 [OUS 83] O USTALOUP A, Syst mes asservis lin aires d ordre fractionnaire, Masson, Paris, S rie Automatique, 1983 [POD 99] P ODLUBNY I, Fractional Differential Equations, Academic Press, Mathematics in Science and Engineering 1998, 1999 [POL 91] P OLACK JD, Time domain solution of Kirchhoff s equation for sound propagation in viscothermal gases: A diffusion process , J Acoustique, vol 4, p 47 67, 1991 [RUS 78] RUSSEL DL, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions , SIAM Rev, vol 20, no 4, p 639 739, 1978 [SAM 87] S AMKO SG, K ILBAS AA, M ARICHEV OI, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, 1987 [SCH 65] S CHWARTZ L, M thodes math matiques pour les sciences physiques, Hermann, Paris, Collection Enseignement des sciences, 1965 [SON 90] S ONTAG ED, Mathematical Control Theory Deterministic Finite Dimensional Systems, Springer-Verlag, Texts in Applied Mathematics 6, 1990 [STA 94] S TAFFANS OJ, Well-posedness and stabilizability of a viscoelastic equation in energy space , Trans Amer Math Soc, vol 345, no 2, p 527 575, 1994 [TAY 96] TAYLOR ME, Partial Differential Equations II: Qualitative Studies of Linear Equations, Springer-Verlag, Applied Mathematical Sciences 116, 1996 [TOR 84] T ORVIK PJ, BAGLEY RL, On the appearance of the fractional derivative in the behavior of real materials , J Appl Mech, vol 51, p 294 298, 1984

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81 Traditional and less traditional questions about fractionals Linear nite impulse response lters enable the design of the well-known moving average (MA) processes The corresponding inverse lters are used to construct autoregressive (AR) processes In this chapter, we study a family of lters receiving growing interest: fractional lters They enable the de nition of fractional processes as well as that of fractional Brownian motion 811 Notes on terminology The word fractional is associated with lters and processes It is both adapted and unadapted On the one hand, it is adapted as it refers to the interesting class of non-integer derivative differential equations, which are constant-coef cient equations p 1 2 with non-integer derivatives of order n , n , , n multiples of a basic fraction [MIL 93, OLD 74] On the other hand, the term is inappropriate because, for most mathematical or ltering issues, power exponents do not consist of fractions: they can be non-rational real numbers, or even complex numbers Nevertheless, as the term is commonly used in the literature, and as it is of no major consequence, we will continue using it 812 Short and long memory Many stochastic models with short memory are known, eg, independent variables, m-dependent variables, certain Markov processes, moving average, most

1JK Pinto, P Rouhiainen, and JW Trailer (1998) Customer-Based Project Success: Exploring a Key to Gaining Competitive Advantage in Project Organizations, Project Management, 4(1), 6 12

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of the autoregressive moving average (ARMA) processes and many linear processes The key consequence of short memory is that it often implies many limit theorems hold, such as the laws of large numbers, central limit theorems, large deviation theorems With long memory, the situation is different It has been more than a century since astronomers rst noticed the existence of empirical series with persistent memory Since then, similar phenomena have been observed, especially in chemistry, hydrology, climatology and economics Such series obviously pose interesting statistical problems and their modeling as well as statistical processing have always been burning issues See [BER 94] for historical examples modeled with fractional Brownian motion Let us mention the existence of two reviews of long-range dependence, one written by Cox [COX 84] and the other by Beran [BER 94] Moreover, a bibliographical guide has been created in [TAQ 92] We focus on second order stationary processes, that is to say assuming the existence of a spectral density f In this case, the habit is taken to characterize long memory by the non-summability of auto-covariance or by the non- niteness of spectral density at the origin However, both phenomena are not equivalent Indeed, there exist fractional ARMA family processes whose spectral density is nite and non-zero, although their autocorrelation is non-summable Furthermore, some authors have recently emphasized how useful spectral density models with singularities away from the origin were for explaining periodicity persistence This is why we de ne long memory as the existence of a frequency 0 in the neighborhood of which spectral density f ( ) behaves as | 0 |2d where d is negative Additionally, the case where 2d is non-integer positive de nes the intermediate memory situation 813 From integer to non-integer powers: lter based sample path design Let us begin with discrete time The well known and often studied ARMA processes are de ned as solutions of recurrence equations whose second member is a simple process: X(t + 1) a0 X(t) a1 X(t 1) + ap X(t p) = b0 (t) b1 (t 1) + bq (t q) where is a white noise iid, innovation of the process By de ning two polynomials and a rational fraction:.

A(z) =

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B(z) =

bj z j ,

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