Ergodicity of nonlinear first order autoregressive models

Abstract

Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfyingX n+1 =f(X n )+σ(X n )ɛ n+1 , wheref, σ are measurable, {ɛ n } are i.i.d. with a (common) positive density,E|ɛ n |>∞. In the special casef(x)/x has limits, α, β asx→−∞ andx→+∞, respectively, it is shown that “α<1, β<1, αβ<1” is sufficient for geometric ergodicity, and that “α<-1, β≤1, αβ≤1” is necessary for recurrence.