On strong limit theorems for general information sources with an application to AEP

Abstract

In this article we prove some strong law of large numbers of arithmetic and geometric mean for a sequence of general information source $\mathbf{X}={\left\{X^n\right\}}_{n\in \mathbb{N}}.$ Under some appropriate conditions, we also obtain a strong limit theorem on harmonic mean of the transition probabilities of X. Finally, as a simple application to information theory, we establish some asymptotic equipartition properties for the generalized entropy density of information sources.

Keywords:

• General information source
• Borel-Cantelli lemma
• geometric mean
• harmonic mean
• entropy density

AMS SUBJECT CLASSIFICATION:

• 60F15
• Secondary: 94A17

Acknowledgments

The authors are grateful to the Referees for their careful work and thoughtful suggestions that have helped improve this article substantially.

Additional information

Funding

Foundation item: Supported by the National Natural Science Foundation of China (No. 11571142), the Natural Science Foundation of Anhui Province (No. 1908085MG227) and the Key Nature Science Research Project of University of An Hui Province (No. KJ2017A547, KJ2017A851, KJ2017A042).