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and the mixing coef cient of maximum correlation by: (A, B) =

.

X L2 (A),Y L2 (B)

corr(X, Y )

.

It is said that a process X is mixing if the sigma-algebras generated by {X(t), t I} and {X(t), t J} have a mixing coef cient which tends towards 0 when d(I, J) tends towards in nity Here, we will use the mixing coef cient:

0 + r(n) = sup |cov(u, v)|, u H , v Hn , Var (u) = Var (v) = 1 s where Hr is the subspace of L2 generated by (X(j), j {r, , s}) Then, (X(n)) is r-mixing if and only if F is a polynomial or if a > 1 and, in the latter case, r(n) = O(an ) If, moreover, (X(n)) is Gaussian, then (X(n)) is strongly mixing if and only if F is a polynomial or if a > 1 and, in this last case, (X(n)) is -mixing with (n) = O(an )

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then, if a > 1, the process (X(n)) is -mixing with (n) = O(a 2n/3 ) According to the values of the parameters a and d, there are thus cases where the process X(n) is: with long memory and not mixing; with short memory and not mixing; with short memory and mixing 833 Parameter estimation In the context of long memory processes, there are two facets to estimation problems The rst is when long memory behaves like a parasitic phenomenon that tends to make traditional results obsolete Examples of this kind are: regression parameters estimation with long memory noise, estimation of the marginal laws of a long memory process and rupture detection in a long memory process The other aspect relates to the estimation of the parameters quantifying the memory length, ie, in fact, spectral density parameters Within this last framework, three types of methods were largely studied, depending on whether the model was completely or partially parameterized:

1) methods related to maximum likelihood, which, by nature, aim at estimating all the parameters of the model; 2) the estimation of the memory exponent d in ( ) = f ( )| | 2d where is the spectral density; 3) the estimation of the autosimilarity parameter in fractional Brownian motion For fractional ARMA processes whose transfer function has the form:

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