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in which the stability property by addition is replaced by the stability property for an extreme value, ie the minimum Weibull s distribution corresponds to this structure This was the beginning of a research trend that would lead to the rediscovery in nance, during the 1990s, of the theory of extreme values,13 which depicts another form of invariance: invariance compared with consideration of the maxima and minima

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13 For the application of the theory of extreme values in nance, see [LON 96, LON 00]

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Scaling, Fractals and Wavelets

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134 Research of dependency and memory of markets 1341 Linear dependence: testing of H-correlative models on returns 13411 Question of dependency of stock market returns The standard model of stock market variations made a hypothesis that returns r(t, ) = ln S(t) ln S(t ) were iid according to a normal variance law 2 The question of validating the independence hypothesis emerged very early in the empirical works dealing with the characterizations of stock market uctuations Generally, dependency between two random variables X and Y is measured by the quantity Cf,g (X, Y ) = E(f (X)g(Y )) E(f (X)]E[g(Y )) and we have the relation: independent X and Y Cf,g (X, Y ) = 0

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The case of f (x) = g(x) = x corresponds to the measurement of usual covariance Other cases include all the (non-linear) possible correlations between variables X and Y Applied to stock market variations, this measure implies that the returns r(t, ) are independent only if we have C(h) = Cf,f ( r(t), r(t+h)) = 0 for any function f ( r(t)) Therefore, studying the independence of stock market variations will pave the way for the analysis of function: C(h) = E f r(t) f r(t + h) E f r(t) E f r(t + h) (1329)

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The chronology of the study merges with different choices made for the de nition of function f ( ) The earliest works (1930-1970) on the veri cation of increment independence were done only on f (x) = x In this case, C(h) becomes the common autocovariance function: C(h) = (h) = E r(t) r(t + h) E r(t) E r(t + h) (1330)

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and the independence of increments corresponds to the invalidity of the linear correlation coef cient In total, the conclusions of initial works proved the absence of a serial autocorrelation and contributed to the formation of a concept of informational ef ciency in stock markets14

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14 See, for example, [CAM 97, TAY 86] for a review of this form of independence and [WAL 96] for the historical formation of the ef ciency concept from initial works

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Research of Scaling Law on Stock Market Variations

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13412 Problem of slow cycles and Mandelbrot s second model However, by the end of the 1970s, certain results contrary to this relation came up in the study of return behaviors in a long-term horizon, which led to tests called long memory By noting by (h) = E( r(t) r(t + h)) E( r(t))E( r(t + h)) the common autocovariance function and (h) = (h)/ (0) the associated autocorrelation function, the standard model of stock market variations implies that (h) must decrease geometrically, ie, (h) cr h with c > 0 However, it seemed that, in some cases, we obtain a hyperbolic decay (h) ch2H 2 with c > 0 and 0 < H < 1, corresponding to a phenomenon of long memory or long dependence This phenomenon of long memory was observed in the 1960s by Adelman [ADE 65] and Granger [GRA 66]; the latter described it as the characteristic of uctuating economic variables Besides, this led Mandelbrot [MAND 65] to rediscover Hurst s law [HUR 51] by introducing the concept of self-similar process which later became [MAND 68] fractional Brownian motion (FBM), whose increments are self-similar with exponent H and autocovariance function (h) = 1 [|h + 1|2H 2|h|2H + |h 1|2H ] Hence, Mandelbrot s model can be 2 quali ed as H-correlative model Mandelbrot called it Joseph s effect with reference to the slow and aperiodic cycles evoked in biblical history concerning Joseph and the uctuations in the Egyptian harvest [MAND 73a] Summers [SUM 86], Fama and French [FAM 88], Poterba and Summers [POT 88] and DeBondt and Thaler [DEB 89] highlighted the phenomena of average return for successive returns, introducing the concept of long-term horizon on markets Although divergent, the interpretations of these autocorrelation phenomena on a long horizon tended to question the hypothesis of common independence and to nd a form of long memory on stock market returns 13413 Introduction of fractional differentiation in econometrics Since the 1970s, econometric limits of ARMA (p, q) and ARIMA (p, d, q) stationary processes in the description of nancial series had progressively led to a generalization of these models by introducing a non-integer differentiation degree 0 < d < 1 with ARFIMA process, which found a great echo in nance in the 2 1980s The fractional differentiation operator d = (1 L)d where is de ned by X(t) = X(t) X(t 1) = (1 L)X(t) and:

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