Geometric ergodicity of nonlinear first order autoregressive models

We consider the ergodic Markov chain satisfying Xn+1 = h(Xn) + (Xn)εn+1, where {εn} is a sequence of independent, identically distributed random variables, considered by Bhattacharya and Lee. After modifications of their sufficient conditions for ergodicity, we give slightly different sufficient conditions for geometric ergodicity on finiteness of an exponential moment of |εn|. Our conditions are weaker than theirs concerning the main part h(Xn) of the Markov chain.