Nonlinear autoregressive processes

Models of the form $X_{n+1}=\lambda (X_{n})+Z_{n+1}$ are considered for time-series {$X_{n}$}, where {$Z_{n}$} is an impulse sequence and $\lambda $ is a nonlinear function. These processes extend the range of behaviour available with linear autoregressive-moving average models. Methods for approximating the stationary distributions of the processes are considered and expressions are found by which the exact moments, joint moments and densities of stationary processes can be obtained. Moments and densities of conditional (predictive) distributions are also found. The results of these methods have been verified by computer simulations and these and other numerical results are given. The extension of the methods obtained to treat multivariate processes of the same form is indicated briefly.