Moderate deviations for randomly perturbed dynamical systems
type: Item
instance of: scholarly article
title: Moderate deviations for randomly perturbed dynamical systems
zbMATH Open document ID: 0961.60042
DOI: 10.1016/S0304-4149(98)00075-1
published in: Stochastic Processes and their Applications
publication date: 2001-01-17T00:00:00Z
review text: Small \(\varepsilon \) asymptotics of a Markov chain defined as a randomly perturbed discrete time dynamical system \[ X_n^{\varepsilon} = f(X_{n-1}^{\varepsilon}) +\varepsilon \eta _n^{\varepsilon} (\omega , X_{n-1}^{\varepsilon}) \] with the same initial condition \(X_0^{\varepsilon} =X_0\in R\) is studied, where \(f\) is a bounded continuous real function, \(\eta ^{\varepsilon}_n\) is measurable for each \(\varepsilon \) and \(n\) and forms an i.i.d. zero mean sequence of random variables for each \(x\). The paper deals with asymptotic analysis, as \(\varepsilon \to 0\), of the centered and normalized process \[ Y^{\varepsilon , \alpha}=\frac {1}{\varepsilon ^{1-\alpha}}[X^{\varepsilon}-X^0],\quad 1/2<\alpha <1. \] Under some conditions the processes \(Y^{\varepsilon , 1/2}\) and \(Y^{\varepsilon ,1}\) obey the central limit theorem and the large deviation principle, respectively. For \(1/2<\alpha <1\), asymptotic behaviour called the moderate deviation principle is shown and explicit rate function is found. An application to a model of density dependent branching process is given.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 1554296
zbMATH Keywords: large deviations
zbMATH Keywords: moderate deviations
zbMATH Keywords: Markov chains
author: Fima Klebaner
author: R. Liptser
MaRDI profile type: Publication
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full work available at URL: s0304-4149(98)00075-1
OpenAlex ID: W1965749111
Wikidata QID: Q127725317
description: scientific article; zbMATH DE number 1554296