Thread: Stationary process and autoregressive model

1. 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).