Non-stationary version of Ergodic Theorem for random dynamical systems (original) (raw)

A Multiplicative Ergodic Theorem for Discontinuous Random Dynamical Systems and Applications

Cornell University - arXiv, 2012

Motivated by studying stochastic systems with non-Gaussian Lévy noise, spectral properties for a type of linear cocycles are considered. These linear cocycles have countable jump discontinuities in time. A multiplicative ergodic theorem is proved for such linear cocycles. Then, the result is illustrated for two linear stochastic systems with general Lévy motions.

On strong ergodicity of iterated function systems, with applications to time series models

Annals of the University of Craiova

We consider so-called iterated function systems, mainly as a mathematical model of (non-linear) autoregressive time series. We apply recent results on Markov chains to iterated function systems and such time series. The goal is to ensure the aperiodic strong ergodicity of Markov chains generated by iterated function systems, respectively of the time series.

ergodic properties of nonhomogeneous markov chains defined on ordered banach spaces with a base

It is known that the Dobrushin's ergodicity coefficient is one of the effective tools to study the behavior of non-homogeneous Markov chains. In the present paper, we define such an ergodicity coefficient of a positive mapping defined on ordered Banach spaces with a base (OBSB), and study its properties. In terms of this coefficient we prove the equivalence uniform and weak ergodicities of homogeneous Markov chains. This extends earlier results obtained in case of von Neumann algebras. Such a result allows to establish a category theorem for uniformly ergodic Markov operators. We find necessary and sufficient conditions for the weak ergodicity of nonhomogeneous discrete Markov chains (NDMC). L-weak ergodicity of NDMC defined on OBSB is also studied. We establish that the chain satisfies L-weak ergodicity if and only if it satisfies a modified Doeblin's condition (D 1-condition). Moreover, some connections between L-weak ergodicity and L-strong ergodicity have been established. Several nontrivial examples of NDMC which satisfy the D 1-condition are provided.

An ergodic theorem for permanents of oblong matrices

We form a sequence of oblong matrices by evaluating an integrable vector-valued function along the orbit of an ergodic dynamical system. We obtain an almost sure asymptotic result for the permanents of those matrices. We also give an application to symmetric means.

From discrete- to continuous-time ergodic theorems

Ergodic Theory and Dynamical Systems, 2012

We introduce methods that allow us to derive continuous-time versions of various discrete-time ergodic theorems. We then illustrate these methods by giving simple proofs and refinements of some known results as well as establishing new results of interest.

Ergodic properties of stationary stable processes

Stochastic Processes and their Applications, 1987

We derive spectral necessary and sufficient conditions for stationary symmetric stable processes to be metrically transitive and mixing. We then consider some important classes of stationary stable processes: Sub-Gaussian stationary processes and stationary stable processes with a harmonic spectral representation are never metrically transitive, the latter in sharp contrast with the Gaussian case. Stable processes with a harmonic spectral representation satisfy a strong law of large numbers even though they are not generally stationary. For doubly stationary stable processes. sulTicient conditions are derived for metric transitivity and mixing, and necessary and sut?icient conditions for ;I strong law of large numbers. AhfS IYYO Strhjec, Cltrssi/icorirm: Primary 6OEO7. 60GlO. 47DI0, 28c)IO stable processes * ergodic theory * stationary processes l spectral representations Research supported by the Air Force Ollke of Scientific Research Contract No. AFOSR F49620 82 c 0009.

On the convergence of the time average for skew-product structure and multiple ergodic system

arXiv: Dynamical Systems, 2017

In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation and observation function are neglected in some measure-theoretical sense. The theorems extend the classical results which have been established for continuous dynamical systems or continuous observation functions. Meanwhile, on the torus mathbbTd\mathbb{T}^{d}mathbbTd with special rotation, we prove the pointwise convergence of multiple ergodic average dispf1Nsumn=0N−1f1(Ralphanx)f2(Ralpha2nx)\disp \f 1 N \sum_{n=0}^{N-1} f_{1}(R_{\alpha}^{n}x)f_{2}(R_{\alpha}^{2n}x)dispf1Nsumn=0N−1f1(Ralphanx)f2(Ralpha2nx) in mathbbTd\mathbb{T}^{d}mathbbTd.