Moderate deviations for randomly perturbed dynamical systems
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.
scientific article; zbMATH DE number 1554296
scientific article