Stationary process and autoregressive model

**akbar** · January 29th 2010, 08:29 AM

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.

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