This paper identifies a class of non-stationary vector ARMA processes with one unit characteristic root that can be used to model multiple time series that stablize together. Analysis of ARMA processes with unit roots is generally more difficult because of their probabilistic roperties. In particular, general theory of law of large numbers and central limit theory do not apply to tandardized sums of the realizations of these processes. However, this paper establishes a weak law of large numbers for a useful class of such processes.