Stability in distribution of randomly perturbed quadratic maps as Markov processes

Iteration of randomly chosen quadratic maps defines a Markov process: Xn+1=.n+1Xn(1.Xn), where .n are i.i.d. with values in the parameter space [0,4] of quadratic maps F.(x)=.x(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 Xn.