Summary.
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).
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Received: 20 March 2002, Revised: 4 December 2002,
JEL Classification Numbers:
C22, D9.
The author wishes to thank Professor Mukul Majumdar and the referees for several useful suggestions.
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Athreya, K.B. Stationary measures for some Markov chain models in ecology and economics. Economic Theory 23, 107–122 (2004) (2003). https://doi.org/10.1007/s00199-002-0352-1
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DOI: https://doi.org/10.1007/s00199-002-0352-1