Summary
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.
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Kateb, D., Seghier, A., Teyssière, G. (2007). Prediction, Orthogonal Polynomials and Toeplitz Matrices. A Fast and Reliable Approximation to the Durbin-Levinson Algorithm. In: Teyssière, G., Kirman, A.P. (eds) Long Memory in Economics. Springer, Berlin, Heidelberg . https://doi.org/10.1007/978-3-540-34625-8_8
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