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Prediction, Orthogonal Polynomials and Toeplitz Matrices. A Fast and Reliable Approximation to the Durbin-Levinson Algorithm

  • Djalil Kateb
  • Abdellatif Seghier
  • Gilles Teyssière

Summary

Let f be a given function on the unit circle such that f(e ) = | 1−e | f 1(e ) 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 ij )0≤i,jN 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.

Keywords

Orthogonal Polynomial Approximation Formula Toeplitz Matrix Stochastic Volatility Model Toeplitz Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Berlin · Heidelberg 2007

Authors and Affiliations

  • Djalil Kateb
    • 1
  • Abdellatif Seghier
    • 2
  • Gilles Teyssière
    • 3
  1. 1.L.M.A.C, Centre de Recherche de RoyallieuUniversité de Technologie de CompiègneCompiègne
  2. 2.Laboratoire de MathématiquesUniversité Paris-SudOrsay Cedex
  3. 3.Statistique Appliquée et de MOdélisation StochastiqueUniversité Paris 1Paris

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