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Math Help - Stationary process and autoregressive model

  1. #1
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    Stationary process and autoregressive model

    Let X_n, n \in \mathbb{Z} be a wide-sense stationary process with spectral measure \rho. Under what conditions on \rho does there exist a wide-sense stationary process Y_n such that:
     X_n=2Y_n-Y_{n-1}-Y_{n+1} \ n\in \mathbb{Z} ?

    Using a usual approach in autoregressive models, one can look for a solution within the Hilbert space of the (X_n)_n in the form: Y_n=\sum_0^{\infty}a_kX_{n-k}. Through identification, we get to the "limit case" with a polynomial of the form: (x-1)^2=0.
    Assuming the X_n have mean zero, we get Y_n=-\sum_0^{\infty}kX_{n-k}, which impose convergence (in L^2) conditions in terms of the spectral measure of the form: \sum_{i,j}\int_{[0,1[}ije^{2i\pi\lambda(j-i)}d\rho(\lambda)<\infty (e.g. \rho is such that the vector \sum_k ke^{-2i\pi\lambda k} has finite norm in L^2([0,1[,\rho))

    Is is possible to do better? (unless there is a better approach to get to a proper condition).

    Thanks for your help.
    Last edited by akbar; January 29th 2010 at 08:35 AM. Reason: additional details
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