| Abstract: | A study of the natures of the processes that are solutions of some finite difference stochastic equations, the right‐hand member of which is a stationary process, is given in the paper. Since the principal application of the present work concerns ARIMA models with or without seasonal variation, these processes are named G‐ARIMA. First, a criterion for a G‐ARIMA process to be stationary is established and some properties of a special class of stationary G‐ARIMA processes are studied. Then, we deduce some conditions for a finite difference stochastic equation to uniquely possess non‐stationary solutions. These have the particular property that their backward shift‐operator may be either a linear or a non‐linear operator, depending on the initial conditions of the solutions. So criteria are established for a non‐stationary G‐ARIMA process to have a bounded linear backward shift operator. Finally, some further properties of the G‐ARIMA processes are given, by comparing them with the broad class of V‐bounded processes. This comparison shows that a non‐stationary process cannot be at the same time G‐ARIMA and V‐bounded. |
