On nonergodicity for nonparametric autoregressive models
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RESEARCH
Open Access
On nonergodicity for nonparametric autoregressive models Mingtian Tang and Yunyan Wang* *
Correspondence: [email protected] School of Science, Jiangxi University of Science and Technology, Ganzhou, 341000, China
Abstract In this paper, we introduce a class of nonlinear time series models with random time delay under random environment, sufficient conditions for nonergodicity of these models are developed. The so-called Markovnization methods are used, that is, proper supplementary variables are added to a non-Markov process, then a new Markov process can be obtained. MSC: 60J05; 60J10; 60K37 Keywords: Markov chains; random delay; random environment; nonergodic
1 Introduction By virtue of their superduper properties, stable (ergodic or recurrent) stochastic processes are very popular among many researchers, so there has been a large literature devoted to the stable (ergodic or recurrent) or even stationary stochastic processes. For instance, Jeantheau [] and Tjøstheim [] established consistency of the estimator they proposed under stationarity and ergodicity conditions (see also [–]). Fernandes and Grammig [] established conditions for the existence of higher-order moments, strict stationarity, geometric ergodicity and β-mixing property with exponential decay. This owes a great deal to the beautiful properties of stable processes, such as an ergodic Markov chain has an invariant probability measure which is finite, a recurrent stochastic process re-visits an arbitrary point in its image an infinite number of times. Just because of this, many researchers often like to target ergodicity or recurrence as their assumptions in their papers or books. However, in this colorful world, lots and lots of phenomena exhibit instability behavior, for example, David [] argued that an important lesson from economic history was that economies exhibited nonergodic behavior along many dimensions. Margolin and Barkai [] indicated that time series of many systems exhibited intermittency, that is to say, at random times the system will switch from state on (or up) to state off (or down) and vice versa. One method to characterize such time series is using time average correlation functions to exhibit a nonergodic behavior. Hence more and more researchers become increasingly interested in these instable processes. Recently, some problems of nonergodic stochastic processes have been studied by many authors. Basawa and Koul [], Basawa and Brockwell [], Basawa and Scott [] and Feigin [] studied asymptotic inference problems for parameters of nonergodic stochastic processes. Budhiraja and Ocone [] proved an asymptotic stability result for discrete © 2013 Tang and Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Tang and Wang Advances in Difference Equations
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