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and: N 2HN 1 VN (w) IN (w) = F (2HN ) almost surely converge when N + towards H1 and:
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(662)
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I(w) =
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a2 (t)w(t) dt
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Moreover: if H2 H1 > 1 , then: 2 N H N H1 and N IN (w) I(w) LogN
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converge in distribution towards a centered random Gaussian variable; 1 if H2 H1 2 , then: E HN H1 and: E IN (w) I(w)
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CN 2(H1 H2 )
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CLog2 (N ) N 2(H1 H2 )
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As regards the estimate of the functional parameter, the preceding theorem can disappoint, which limits itself to proposing an estimate of integrals of a2 against the weight functions w To rebuild a pointwise estimator of a(t) starting from these integrals, a general method will be found in [IST 96] To understand the convergence speeds determined by Theorem 67, it is necessary to know that the convergences of the estimators reveal two types of error One comes from the central limit theorem and intervenes in the estimate of the rst-order parameters; we will call it stochastic error On the other hand, the second-order disturbance in the lter de ning the ltered white noise creates a distortion If H2 H1 > 1 , stochastic error is dominant over 2 distortion Otherwise, the convergence speed is imposed by distortion The estimate of the second-order factors (b, H2 ) is more dif cult because it necessitates that we build functions that do not depend asymptotically on the rst-order factors An example of such a functional is given by: V N 22H1 1 VN
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The intervention of the factor 22H1 1 is necessary to compensate for the in uence of the rst-order parameters exactly On the other hand, it must be estimated and for this we will use the convergence of: V N2
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N +
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= 22H1 1
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which is suf ciently rapid for the compensation to always take An estimator of the parameter H2 is thus obtained THEOREM 68 If the function: WN = V N
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V N2
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(663)
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the estimator: H2 (N ) = VN 2 /2 WN/2 1 1 log2 + log2 2 2 VN 2 WN (664)
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converges as towards H2 when N + 6522 Identi cation of elliptic Gaussian random processes Although it is possible to directly identify the symbol of an elliptic Gaussian random process of the form: (t, ) = a(t)| | 2 +H1 + b(t)| | 2 +H2 + p( )
(665)
when 0 < H2 < H1 < 1, for a and b two strictly positive C 1 functions, and for 1 and p( ) = 0 if | | > 2 (see p a c function such that p( ) = 1 if | | [BEN 94]); a comparison carried out in [BEN 97a] between ltered white noises and elliptic Gaussian random processes makes it possible to obtain the result more easily Let us make several comments on the symbols which we identify The symbols of form (665) verify Hypothesis 63 (H1 ) In particular, function p was introduced so that the elliptic inequality of order H1 in = 0 would be satis ed In fact, (665) should be understood as an expansion in the fractional power in high frequency (| | + ) of a general symbol The identi cation of the symbol of an elliptic Gaussian random process X comes from the comparison of X with the ltered white noise: Yt = e it 1 W (d ) (t, ) (666)
This explains that the order of the powers for an elliptic Gaussian random process is reversed compared to that which we have for a ltered white noise The results of identi cation for the elliptic Gaussian processes can be summarized by recalling the following theorem
Locally Self-similar Fields
THEOREM 69 If X is an elliptic Gaussian random process of the symbol verifying (665) and: sup 0, then the estimators: VN/2 1 log2 HN = +1 2 VN N 2HN 1 VN (w) JN (w) = F (2HN ) for w C 2 [0, 1] with support included in ]0, 1[ and: (H2 )N = VN 2 /2 WN/2 1 1 + log2 log2 2 2 VN 2 WN (669)