Stationary measures for some Markov chain models in ecology and economics

Abstract

Let $F \equiv \{f : f : [0, \infty) \rightarrow [0, \infty), f (0) = 0, f$ continuous, $\lim\limits_{x \downarrow 0} \frac{f(x)}{x} = C$ exists in $(0, \infty), 0 < g (x) \equiv \frac{f(x)}{C x} < 1$ for x in $(0, \infty)\}$ . Let $\{f_j\}_{j \geq 1}$ be an i.i.d. sequence from F and X 0 be a nonnegative random variable independent of $\{f_j\}_{j \geq 1}$ . Let $\{X_n\}_{n \geq 0}$ be the Markov chain generated by the iteration of random maps $\{f_j\}_{j \geq 1}$ by $X_{n + 1} = f_{n + 1} (X_n), n \geq 0$ . 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).