The determination of periodic stationarity conditions for periodic autoregressive moving average (PARMA) processes is a prerequisite to their analysis. Means of obtaining these conditions in analytically simple forms are sought. It is shown that periodic stationarity conditions for univariate and multivariate PARMA processes can always be reduced to eigenvalue problems, which are computationally and analytically easier to deal with. Two different lumpings of the periodic process are considered along this line. The first is the common w-span lumping over all w periods. The second lumping considered is the p-span lumping of the pth order periodic autoregressive process over p periods, which is based on a recently introduced lumping technique. It is shown that p-span lumping may yield the periodic stationarity conditions in an analytically simpler form as compared to w-span lumping when p¿ 1997 American Geophysical Union