[Paper Review] Dimension of the space of invariant finitely additive measures of general Markov chains and their ergodic properties

Alexander I. Zhdanok|arXiv (Cornell University)|Apr 9, 2018

Mathematical Dynamics and Fractals3 citations

TL;DR

This paper establishes a full inversion of a prior theorem on Markov chains by proving that if the space of invariant purely finitely additive measures is finite-dimensional (under additional conditions), then the Markov chain satisfies Doob-Dobrushin quasicompactness conditions. The work extends earlier partial results (valid only for dimension one) and demonstrates that purely finitely additive invariant measures are essential for understanding ergodic properties and asymptotic behavior, even when countably additive invariant measures are absent.

ABSTRACT

General Markov chains with a countably additive transition probability in arbitrary phase space are considered. Markov operators extend from the space of countably additive measures to the space of finitely additive measures. In the author's papers a theorem was earlier proved that if all invariant finitely additive measures of a Markov chain are countably additive, i.e. there are no invariant purely finitely additive measures, then their subspace is finite-dimensional and the Markov chain satisfies the Doob-Doeblin quasicompactness conditions. In the same paper, a partial inversion of this theorem was proved with the dimension "one". In this paper we prove the inversion of this assertion for any finite dimensionality, but under certain additional conditions. The ergodic consequences are given. Examples and methods for studying their asymptotics with the aid of invariant purely finitely additive measures are given.

Motivation & Objective

• To extend a previously proven partial inversion of a theorem linking the absence of purely finitely additive invariant measures to quasicompactness in general Markov chains.
• To establish a full inversion of this theorem for any finite dimension of the space of invariant purely finitely additive measures.
• To demonstrate the critical role of purely finitely additive invariant measures in characterizing the asymptotic and ergodic properties of Markov chains.
• To provide a framework and examples for studying the asymptotics of Markov chains using purely finitely additive invariant measures.

Proposed method

• Extends the Markov operator A from the space of countably additive measures (ca) to the space of finitely additive measures (ba), preserving its action on bounded measurable functions.
• Analyzes the structure of the space of invariant finitely additive measures, decomposing it into countably additive and purely finitely additive components.
• Applies the theory of finitely additive measures, particularly the concept of purely finitely additive (pure charge) measures, which are discontinuous with respect to countable additivity.
• Uses the condition that the subspace of invariant purely finitely additive measures is finite-dimensional, combined with additional technical assumptions, to derive quasicompactness.
• Employs examples on the interval [0,1] to illustrate the behavior and asymptotic analysis of chains using purely finitely additive invariant measures.
• Leverages results from prior works by the author (e.g., [4], [5]) and foundational texts on operator theory and measure theory (e.g., [1], [2], [3]) to build the theoretical framework.

Experimental results

Research questions

• RQ1 Under what conditions does the finite-dimensionality of the space of invariant purely finitely additive measures imply quasicompactness of the Markov operator?
• RQ2 How can the asymptotic behavior of a Markov chain be characterized when no countably additive invariant measures exist, but purely finitely additive ones do?
• RQ3 What is the role of purely finitely additive invariant measures in determining ergodic properties of general Markov chains?
• RQ4 How can the asymptotics of such chains be systematically studied using purely finitely additive invariant measures?

Key findings

• The paper proves a full inversion of a prior theorem: if the space of invariant purely finitely additive measures is finite-dimensional and certain additional conditions are met, then the Markov chain satisfies the Doob-Dobrushin quasicompactness conditions.
• The result generalizes a previous partial inversion that was only valid for the case of dimension one.
• The study shows that purely finitely additive invariant measures are crucial for understanding the asymptotic behavior of Markov chains, even when countably additive invariant measures are absent.
• Examples are constructed on the interval [0,1] to illustrate how purely finitely additive invariant measures can be used to analyze the asymptotics of Markov chains.
• The paper establishes that the absence of purely finitely additive invariant measures is a strong indicator of maximal ergodicity, though this is considered a degenerate case rather than a typical one.
• The framework developed allows for a deeper analysis of chains where standard ergodic theory based on countably additive measures fails.