Let , be a wide-sense stationary process with spectral measure . Under what conditions on does there exist a wide-sense stationary process such that:

?

Using a usual approach in autoregressive models, one can look for a solution within the Hilbert space of the in the form: . Through identification, we get to the "limit case" with a polynomial of the form: .

Assuming the have mean zero, we get , which impose convergence (in ) conditions in terms of the spectral measure of the form: (e.g. is such that the vector has finite norm in )

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

Thanks for your help.