Stability in distribution of randomly perturbed quadratic maps as Markov processes

Abstract

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