ON THE QUASI-STATIONARY DISTRIBUTION FOR SOME RANDOMLY PERTURBED TRANSFORMATIONS OF AN INTERVAL

We consider a Markov chain X-n(epsilon) obtained by adding small noise to a discrete time dynamical system and study the chain's quasi-stati onary distribution (qsd). The dynamics are given by iterating a functi on f: I --> I for some interval I when f has finitely many fixed point s, some stable and some unstable. We show that under some conditions t he quasi-stationary distribution of the chain concentrates around the stable fixed points when epsilon --> 0. As a corollary, we obtain the result for the case when f has a single attracting cycle and perhaps r epelling cycles and fixed points. In this case, the quasi-stationary d istribution concentrates on the attracting cycle. The result applies t o the model of population dependent branching processes with periodic conditional mean function.