Regular multidimensional stationary time series

The aim of this article is to give a simpler, more usable sufficient and necessary condition to the regularity of generic weakly stationary time series. Also, this condition is used to show how a regular process can be approximated by a lower rank 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.