Abstract
We consider a positive contraction P defined on a probability space \((X,\Sigma ,m)\) and assume \(P{\textbf{1}}_X={\textbf{1}}_X\) and \(P^*{\textbf{1}}_X={\textbf{1}}_X\) (i.e., P is a bi-stochastic Markov operator). By virtue of the zero-two law, there exists a sufficient condition for the asymptotic periodicity of P. In this paper, we give some conditions between the convergence almost everywhere, the zero-two law, and the asymptotic periodicity for P.
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Acknowledgements
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. The author has declared no conflict of interest. Yukiko Iwata was involved in the conceptualisation of the research, conducted the modelling analysis and wrote the first draft manuscript. All authors interpreted the findings, read and approved the final manuscript. Finally, the author would like to thank T. Honda for his very helpful comments.
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The author, YI (1) made substantial contributions to the study concept or proofs; (2) drafted the manuscript or revised it critically for important intellectual content; (3) approved the final version of the manuscript to be published; (4) agreed to be accountable for all aspects of the work.
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Iwata, Y. Almost everywhere convergence of the iterates of a bi-stochastic Markov operator. Positivity 27, 60 (2023). https://doi.org/10.1007/s11117-023-01010-7
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DOI: https://doi.org/10.1007/s11117-023-01010-7