Strong laws of large numbers for the mth-order asymptotic odd–even Markov chains indexed by an m-rooted Cayley tree

Abstract

In this article, we will study the strong laws of large numbers and asymptotic equipartition property (AEP) for mth-order asymptotic odd–even Markov chains indexed by an m-rooted Cayley tree. First, the definition of mth-order asymptotic odd–even Markov chains indexed by an m-rooted Cayley tree is introduced, then the strong limit theorem for this Markov chains is established. Next, the strong laws of large numbers for the frequencies of ordered couple of states for mth-order asymptotic odd–even Markov chains indexed by an m-rooted Cayley tree are obtained. Finally, we prove the AEP for this Markov chains.

KEYWORDS:

• Asymptotic equipartition property
• m-Rooted Cayley tree
• mth-Order asymptotic odd–even Markov chain
• Strong law of large numbers

MATHEMATICS SUBJECT CLASSIFICATION:

• 60F15
• 60J10

Acknowledgments

We would like to thank the reviewers of this article for their most value comments and suggestion.

Funding

This work is supported by the National Natural Science Foundation of China (11601191, 11571142), Research Foundation for Advanced Talents of Jiangsu University (11JDG116), Statistic Application Research Base of The Education Department of Jiangsu Province, and The Key Discipline of Statistic of Jiangsu University in 2014.