On a theorem of Dubins and Freedman

Under a notion of "splitting" the existence of a unique invariant probabili ty, and a geometric rate of convergence to it in an appropriate metric, are established for Markov processes on a general state space S generated by i terations of i.i.d. maps on S. As corollaries we derive extensions of earli er results of Dubins and Freedman;((17)) Yahav;((30)) and BhattacharSia and Lee((6)) for monotone maps. The general theorem applies in other contexts as well. It is also shown that the Dubins-Freedman result on the "necessity " of splitting in the case of increasing maps does not hold for decreasing maps, although the sufficiency part holds for both. In addition, the asympt otic stationarity of the process generated by i.i.d, nondecreasing maps is established without the requirement of continuity. Finally, the theory is a pplied to the random iteration of two (nonmonotone) quadratic maps each wit h two repelling fixed points and an attractive period-two orbit.