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i hence, to point 2n , we allotted a mass proportional to 2 n(H+d) , since the proportionality coef cient is equal to the volume of B The contribution of these masses at scale n is proportional to 2 nH per volume unit
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5132 Complete coding If we de ne a Cantor set on [0, 1[ only, the resulting set is not stable by dilation of a factor 3 We must de ne this set on R to make it stable by dilation of any unspeci ed power of 3 In the same way, the measure M0 de ned above cannot be stable by dilation of a factor 2 However, the approach used as for Cantor sets can be adopted We outline it only brie y For any n 0, we de ne the measure Mn par Mn (dx) = 2 nH (M0 (x + 2 n ) M0 (x))(dx) As distributions, this sequence Mn converges slightly towards M Thus, we verify that M is semi-self-similar for the multiplicative sub-group of powers of 2 It is also stationary Let us note that our construction seems to ascribe a speci c role with the number 2 This is not the case and we can replace 2 by any b > 0 in (55) We then obtain the semi-self-similar measure for the multiplicative sub-group of powers of b 514 Weierstrass functions With Weierstrass functions, we have a deterministic distribution model of mass with properties analog to equation (54) If b > 1 and 0 < H < 1, for x R, Weierstrass functions Wb,H are de ned as (see [WEI 72]): Wb,H (x) =
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We can easily verify the semi-self-similar property: Wb,H (bx) = bH Wb,H (x) We should note that the preceding constructions, intended to expand Cantor sets and renormalized sums of Poisson measures on R, have their match on Weierstrass functions by writing:
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The question that naturally arises is: are there probabilistic models which are self-similar and stationary The traditional answer is positive, provided the stationarity condition is replaced by a stationary of the increments condition Therefore, we proceed with the introduction of self-similar stochastic processes whose increments are stationary
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515 Renormalization of sums of random variables In this section, we present the results of Lamperti s article [LAM 62] on self-similar process obtained as renormalization limits of other stochastic processes Let us rst recall Lamperti s de nition of a semi-stable 1 stochastic process DEFINITION 51 A stochastic process X(x), with x R, is called semi-stable if, for any a > 0, there is a renormalization function b(a) > 0 such that: X(ax), x R = b(a)X(x), x R When the function b(a) is of the form aH , the process X is called self-similar Lamperti s fundamental result shows that the possible choices for the renormalization function b(a) is actually limited THEOREM 51 ([LAM 62, Theorem 1, p 63]) Any stochastic semi-stable process is self-similar From now on, we must note that this result is not in contradiction with the existence of locally self-similar2 processes (see [BEN 98]) Let X and Y be two real stochastic processes indexed by Rd If we assume hypothesis R, there exists a function f : R+ R+ such that: lim X(ax) , x Rd f (a) = Y (x), x Rd
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Moreover, let us recall that a function L is a slowly varying function if, for any y > 0, we obtain limx + L(xy) = 1 L(x) THEOREM 52 ([LAM 62, Theorem 2, p 64]) Let X and Y be two stochastic processes such that there is a function f for which the hypothesis R is satis ed Then, f necessarily has the following structure, with H > 0: f (a) = aH L(a) where L is a slowly varying function
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1 Not to be confused with the de nition of a stable process 2 These processes are presented in 6
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