Some generalized strong limit theorems for Markov chains in bi-infinite random environments

Abstract

In this paper, we study a strong limit theorem of delayed sums for Markov chains in bi-infinite environments by constructing a sequence of random variables with parameters. As a corollary, we obtain some strong limit properties including the generalized conditional relative entropy for Markov chains in bi-infinite random environment. The results which we obtained generalize the results of Liu et al. (2015).

KEYWORDS:

• Random environments
• Markov chains
• conditional relative entropy
• delayed sums

MATHEMATICS SUBJECT CLASSIFICATION (2000)::

• 60F15
• 60J10

Acknowledgments

Z. Y. Shi acknowledges partial support from the National Natural Science Foundation of China (11601191, 11971197, 11571142), Youth talent cultivation project of Jiangsu University and Young science and technology talents lifting project of Jiangsu association for science and technology. Y. Chen acknowledges partial support from the Humanities and Social Sciences Foundation of the Ministry of Education of China (No. 18YJC910004), Postdoctoral Research Project of Zhejiang Province, and First Class Discipline of Zhejiang—A (Zhejiang Gongshang University—Statistics) and Postdoctoral Training Program of Zhejiang Gongshang University.