  The aim of this paper is to give a simpler, more usable sufficient condition
to the regularity of generic weakly stationary time series. Also, this
condition is used to show how regular processes satisfying these sufficient
conditions can be approximated by a lower rank \emph{regular} process. The
relevance of these issues is shown by the ever increasing presence of
high-dimensional data in many fields lately, and because of this, low rank
processes and low rank approximations are becoming more important. Moreover,
regular processes are the ones which are completely influenced by random
innovations, so they are primary targets both in the theory and applications.
