CLOSURE OF LINEAR-PROCESSES

We consider the sets of moving-average and autoregressive processes an d study their closures under the Mallows metric and the total variatio n convergence on finite dimensional distributions. These closures are unexpectedly large, containing nonergodic processes which are Poisson sums of i.i.d. copies From a stationary process. The presence of these nonergodic Poisson sum processes has immediate implications. In parti cular, identifiability of the hypothesis of linearity of a process is in question. A discussion of some of these issues for the set of movin g-average processes has already been given without proof in Bickel and Buhlmann.((2)) We establish here the precise mathematical arguments a nd present some additional extensions: results about the closure of au toregressive processes and natural sub-sets of moving-average and auto regressive processes which are closed.