ERGODICITY OF NONLINEAR FIRST-ORDER AUTOREGRESSIVE MODELS

Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfying X(n+1) = f(X(n)) +sigma(X(n))epsilon(n+1), where f, sigma are measurable, {epsilon(n)} are i.i.d. with a (common) positive density, E \epsilon(n)\ < infinity. In the special case f(x)/x has limits alpha, beta as x --> infinity and x --> + infinity, respectively, it is shown that ''alpha < 1, beta < 1, alpha beta < 1'' is sufficient for geometric ergodicity, and that ''alpha less than or equal to 1, beta less than or equal to 1, alpha beta less than or equal to 1'' is necessary for recurrence.