Self-similar Processes

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where the function is the denominator of (512) or (513) From this, we deduce the following decomposition: X(x) =

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d 2 jH

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where the

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( )

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verify the following stationarity properties For any j:

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j+ ,k,u = j,k,u L L

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r

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(515)

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j,k+r,u = j,k,u

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This property can be compared with that given in [CAM 95] (for second order processes) or in [AVE 98] (general case) for the wavelet coef cients of self-similar processes with stationary increments 535 Process subordinated to Brownian measure Let ( , F, P) be a probabilized space and Wd the Brownian measure on L2 (Rd ) The space L2 ( , F, P) is characterized by its decomposition in chaos [NEV 68] Let us brie y recall this theory Let n be the symmetric group of order n For any function of n variables, we de ne: def 1 f (x (1) , , x (n) ) f n (x1 , , xn ) = n!

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Let us also de ne the symmetric stochastic measure of order n, ie (n) Wd (dx1 , , dxn ) on L2 ((Rd )n ) by: Wd (A1 An ) = Wd (A1 ) Wd (An ) where any two Ai are disjoint In addition, it is imposed that the expectation (n) of Wd (f1 fn ) is always zero As an example, with n = 2, we obtain 2 2 2 Wd (f g) = Wd (f )Wd (g) f, g : Wd (f ) = Wd (f n )

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(n) def (n) (n) def

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The following properties are established: E Wd (f ), Wd

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(n) (m)

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(g) = n,m f, g 0, with:

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For any F L2 ( , F, P), there exists a sequence fn L2 ((Rd )n ), n F =

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Wd (fn )

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and this decomposition is unique Moreover, we have EF = Wd

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Scaling, Fractals and Wavelets

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THEOREM 516 Let 0 < H < 1: process Y n de ned by: Y n (x1 , , xn ) =

def Rd

eix 1 | |

dn 2 +H

n TF (Wd )(d )

is self-similar (of order H), with stationary increments; process X n de ned by: X n (x) = Y n (x, , x) is self-similar (of order H), with stationary increments; if an is a summable square sequence, process X de ned by: X(x) =

def n 0 def

an X n (x)

is self-similar (of order H), with stationary increments This theorem shows how dif cult a general classi cation of self-similar processes with stationary increments can be Let us note, moreover, that we considered the elliptic case only We could, for example, also think of combinations of hyperbolic and elliptic cases This dif culty is clearly indicated by Dobrushin in the issues raised in the comments of [DOB 79, Theorem 62, p 24] 54 Regularity and long-range dependence 541 Introduction As opposed to what the title of this section may suggest, we will address this question only by means of ltered white noise and for the Gaussian case Despite its restricted character, this class of examples allows us to question the connection between the regularity of trajectories on the one hand and the long-range correlation5 on the other hand To begin with, let us once again consider fractional Brownian motion BH , with parameter H The sample paths of BH are H lderian with exponent h (as), for any

5 The analysis of decorrelation and mixing process properties, is an already old subject (see, for example, [DOU 94, IBR 78])

Self-similar Processes

h < H, but are not H lderian with exponent H (as) In addition, we can verify that, for > 0, k N: E BH ( ) BH (k + 1) BH (k ) =c | |2H |k|2(1 H)

The decrease, with respect to lag , of the correlation of the increments of X is slow It is often incorrectly admitted that the H lderian character and the slow decrease of the correlation of the increments are tied together 542 Two examples 5421 A signal plus noise model Let H and K be such that 0 < K < H < 1 Let S1 and S2 , be two functions on the sphere d 1 with values in [a, b], with 0 < a b < + Then, let us consider the process X de ned by: X(x) =