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Thread: Stationary process and autoregressive model

  1. #1
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    Stationary process and autoregressive model

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

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

    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; Jan 29th 2010 at 08:35 AM. Reason: additional details
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