Stationary measures for some Markov chain models in ecology and economics

Abstract

Let F≡ {f:f:[0, ∞) → [0, ∞), f(0) =0,f continuous,lim _x↓0 f(x)/x=C exists in (0,∞), 0 < g(x) ≡ f(x)/Cx <1 for x in (0,∞). Let {fj}_j≥1 be an i.i.d. sequence from F and X₀ be a nonnegative random variable independent of {fj}_j≥1. Let {X_n}_n≥0 be the Markov chain generated by the iteration of random maps {fj}_j≥1by X_n+1=f_n+1(X_n), n≥0. Such Markov chains arise in population ecology and growth models in economics. This paper studies the existence of nondegenerate stationary measures for {X_n}. A set of necessary conditions and two sets of sufficient conditions are provided. There are some convergence results also. The present paper is a generalization of the work on random logistics maps by Athreya and Dai (2000).