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The Generalized Entropy Ergodic Theorem for Nonhomogeneous Bifurcating Markov Chains Indexed by a Binary Tree.
Zhiyan Shi1, Zhongzhi Wang2, Pingping Zhong1
1School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013 China.
Journal of Theoretical Probability
|August 9, 2021
Summary
This study establishes a generalized entropy ergodic theorem for nonhomogeneous bifurcating Markov chains. It proves a strong law of large numbers for state frequencies and generalizes existing results.
Area of Science:
- Probability Theory
- Stochastic Processes
- Information Theory
Background:
- Ergodic theorems are fundamental in understanding the long-term behavior of stochastic processes.
- Bifurcating Markov chains model systems that split over time, common in fields like population dynamics and branching processes.
- Generalized entropy provides a flexible framework for analyzing information and uncertainty.
Purpose of the Study:
- To establish the generalized entropy ergodic theorem for a specific class of stochastic processes.
- To extend the study of ergodic properties to nonhomogeneous bifurcating Markov chains indexed by a binary tree.
- To generalize and unify existing results in the field.
Main Methods:
- Construction of a specialized class of random variables with a parameter and a mean of one.
- Application of the Borel-Cantelli lemma to establish a strong limit theorem for delayed sums.
- Proof of the strong law of large numbers for state frequencies of delayed sums.
- Derivation of the generalized entropy ergodic theorem.
Main Results:
- A strong limit theorem for delayed sums of bivariate functions of the chains was established.
- The strong law of large numbers for the frequencies of occurrence of states of delayed sums was proven.
- The generalized entropy ergodic theorem was successfully derived for the studied chains.
Conclusions:
- The study successfully extends ergodic theory to nonhomogeneous bifurcating Markov chains.
- The findings provide a theoretical foundation for analyzing complex branching processes.
- The generalized results offer a unified perspective on existing theorems in the area.
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