Stability in distribution of randomly perturbed quadratic maps as Markov processes

Iteration of randomly chosen quadratic maps defines a Markov process: Xn+1 = epsilon(n+1) X-n(1 - X-n), where epsilon(n) are i.i.d. with values in the parameter space [0, 4] of quadratic maps F-theta(x) = thetax(1 - x). Its study is of significance as an important Markov model, with applications to problems of optimization under uncertainty arising in economics. In this article a broad criterion is established for positive Harris recurrence of X-n.