Ergodicity of nonlinear first order autoregressive models

Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfyingX_n+1 =f(X_n)+(X_n)_n+1, wheref, are measurable, {_n} are i.i.d. with a (common) positive density,E|_n|>. In the special casef(x)/x has limits, , asx– andx+, respectively, it is shown that <1, <1, <1 is sufficient for geometric ergodicity, and that <-1, 1, 1 is necessary for recurrence.