We consider autoregressive neural network (AR-NN) processes driven by additive noise and demonstrate that the characteristic roots of the shortcuts—the standard conditions from linear time-series analysis—determine the stochastic behavior of the overall AR-NN process. If all the characteristic roots are outside the unit circle, then the process is ergodic and stationary. If at least one characteristic root lies inside the unit circle, then the process is transient. AR-NN processes with characteristic roots lying on the unit circle exhibit either ergodic, random walk, or transient behavior. We also analyze the class of integrated AR-NN (ARI-NN) processes and show that a standardized ARI-NN process “converges” to a Wiener process. Finally, least-squares estimation (training) of the stationary models and testing for nonstationarity is discussed. The estimators are shown to be consistent, and expressions on the limiting distributions are given.