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Erschienen in:
2007 | OriginalPaper | Buchkapitel
Prediction, Orthogonal Polynomials and Toeplitz Matrices. A Fast and Reliable Approximation to the Durbin-Levinson Algorithm
verfasst von : Djalil Kateb, Abdellatif Seghier, Gilles Teyssière
Erschienen in: Long Memory in Economics
Verlag: Springer Berlin Heidelberg
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Let
f
be a given function on the unit circle such that
f
(
e
iθ
) = | 1−
e
iθ
|
2α
f
1
(
e
iθ
) with | α |< 1/2 and
f
1
a strictly positive function that will be supposed to be sufficiently smooth. We give the asymptotic behavior of the first column of the inverse of
T
N
(
f
), the (
N
+1) × (
N
+ 1) Toeplitz matrix with elements (
f
i
−
j
)
0≤
i,j
≤
N
where
$$ f_k = \tfrac{1} {{2\pi }}\int_0^{2\pi } {f(e^{ - i\theta } )e^{ - ik\theta } d\theta } $$
. We shall compare our numerical results with those given by the Durbin-Levinson algorithm, with particular emphasis on problems of predicting either stationary stochastic long-range dependent processes, or processes with a long-range dependent component.