Journal of Statistical Physics, Vol. 74, Nos. 1/2, 1994. p. 91-109.
Components which are placed in a finite or infinite space have integer numbers as possible states. They interact in a discrete time in a local deterministic way, in addition to which all the components' states are incremented at every time step by independent identically distributed random variables. We assume that the deterministic interaction function is translation-invariant and monotonic and that its values are between the minimum and the maximum of its arguments. Theorems 1 and 2 (based on propositions which we give in a separate Part II), give sufficient conditions for a system to have an invariant distribution or a bounded mean. Other statements, proved herein, provide background for them by giving conditions when a system has no invariant distribution or the mean of its components' states tends to infinity. All our main results use one and the same geometrical criterion.