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February 1998 On the quasi-stationary distribution for some randomly perturbed transformations of an interval
Fima C. Klebaner, Justin Lazar, Ofer Zeitouni
Ann. Appl. Probab. 8(1): 300-315 (February 1998). DOI: 10.1214/aoap/1027961045

Abstract

We consider a Markov chain $X_n^{\varepsilon}$ obtained by adding small noise to a discrete time dynamical system and study the chain's quasi-stationary distribution (qsd). The dynamics are given by iterating a function $f:I \to I$ for some interval I when f has finitely many fixed points, some stable and some unstable. We show that under some conditions the quasi-stationary distribution of the chain concentrates around the stable fixed points when $\varepsilon \to 0$. As a corollary, we obtain the result for the case when f has a single attracting cycle and perhaps repelling cycles and fixed points. In this case, the quasi-stationary distribution concentrates on the attracting cycle. The result applies to the model of population dependent branching processes with periodic conditional mean function.

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Fima C. Klebaner. Justin Lazar. Ofer Zeitouni. "On the quasi-stationary distribution for some randomly perturbed transformations of an interval." Ann. Appl. Probab. 8 (1) 300 - 315, February 1998. https://doi.org/10.1214/aoap/1027961045

Information

Published: February 1998
First available in Project Euclid: 29 July 2002

zbMATH: 0942.60070
MathSciNet: MR1620378
Digital Object Identifier: 10.1214/aoap/1027961045

Subjects:
Primary: 60F10 , 60J80

Keywords: branching systems , large deviations , logistic map , quasi-stationary distribution

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.8 • No. 1 • February 1998
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