On the Quasi-stationary distribution for some randomly perturbed transformations of an interval

Fima C. Klebaner, Justin Lazar, Ofer Zeitouni

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8 Citations (Scopus)

Abstract

We consider a Markov chain Xε n 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 → 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 ε → 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.

Original languageEnglish
Pages (from-to)300-315
Number of pages16
JournalAnnals of Applied Probability
Volume8
Issue number1
DOIs
Publication statusPublished - 1 Jan 1998
Externally publishedYes

Keywords

  • Branching systems
  • Large deviations
  • Logistic map
  • Quasi-stationary distribution

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