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

  • Djalil Kateb
  • Abdellatif Seghier
  • Gilles Teyssière


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.


Orthogonal Polynomial Approximation Formula Toeplitz Matrix Stochastic Volatility Model Toeplitz Matrice 
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© 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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