A note on the ergodicity of non-linear autoregressive model

We examine the Markov chain Xt = [Phi](Xt - 1) + [var epsilon]tb, where Xt = (xt, ..., xt - p + 1)[tau], B = (1, 0, ..., 0)[tau]. Under some appropriate conditions on [Phi], we show the ergodicity for {Xt} when E[var epsilon]t2 is suitable small, and the geometric ergodicity when Ee[var epsilon]t is suitably small.