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Stationary measures for some Markov chain models in ecology and economics

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  • Krishna Athreya

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). Copyright Springer-Verlag Berlin/Heidelberg 2003

Suggested Citation

  • Krishna Athreya, 2003. "Stationary measures for some Markov chain models in ecology and economics," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 23(1), pages 107-122, December.
  • Handle: RePEc:spr:joecth:v:23:y:2003:i:1:p:107-122
    DOI: 10.1007/s00199-002-0352-1
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    Citations

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    Cited by:

    1. Lars J. Olson & Santanu Roy, 2006. "Theory of Stochastic Optimal Economic Growth," Springer Books, in: Rose-Anne Dana & Cuong Le Van & Tapan Mitra & Kazuo Nishimura (ed.), Handbook on Optimal Growth 1, chapter 11, pages 297-335, Springer.
    2. Collamore, Jeffrey F. & Vidyashankar, Anand N., 2013. "Tail estimates for stochastic fixed point equations via nonlinear renewal theory," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3378-3429.
    3. Mitra, Tapan & Roy, Santanu, 2007. "On the possibility of extinction in a class of Markov processes in economics," Journal of Mathematical Economics, Elsevier, vol. 43(7-8), pages 842-854, September.

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