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Recurrent neural networks (RNNs) have been widely used to model nonlinear dynamic systems using time-series data. While the training error of neural networks can be rendered sufficiently small in many cases, there is a lack of a general framework to guide construction and determine the generalization accuracy of RNN models to be used in model predictive control systems. In this work, we employ statistical machine learning theory to develop a methodological framework of generalization error bounds for RNNs. The RNN models are then utilized to predict state evolution in model predictive controllers (MPC), under which closed-loop stability is established in a probabilistic manner. A nonlinear chemical process example is used to investigate the impact of training sample size, RNN depth, width, and input time length on the generalization error, along with the analyses of probabilistic closed-loop stability through the closed-loop simulations under Lyapunov-based MPC.