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Ergodicity of p−majorizing nonlinear Markov operators on the finite dimensional space.
- Source :
-
Linear Algebra & its Applications . Oct2019, Vol. 578, p53-74. 22p. - Publication Year :
- 2019
-
Abstract
- A nonlinear Markov chain is a discrete time stochastic process whose transitions may depend on both the current state and the current distribution of the process. The nonlinear Markov chain over a finite state space can be identified by a continuous mapping (the so-called nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex) of the finite state space and by a family of transition matrices depending on occupation probability distributions of states. In this paper, we introduce a notion of Dobrushin's ergodicity coefficients for stochastic hypermatrices and provide a criterion for the contraction nonlinear Markov operator by means of Dobrushin's ergodicity coefficients. We also introduce a notion of p − majorizing nonlinear Markov operators associated with stochastic hypermatrices and provide a criterion for strong ergodicity of such kind of operator. We show that the p − majorizing nonlinear Markov operators associated with scrambling , Sarymsakov , and Wolfowitz stochastic hypermatrices are strongly ergodic. These classes of p − majorizing nonlinear Markov operators assure an existence of a residual set of strongly ergodic nonlinear Markov operators which are not contractions. Some supporting examples are also provided. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 578
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 137510243
- Full Text :
- https://doi.org/10.1016/j.laa.2019.05.011