Finite ergodic components for upper probabilities

TL;DR

This paper introduces finite ergodic components for upper probabilities, establishing their properties and equivalences with eigenvalues, law of large numbers, and asymptotic independence, extending classical ergodic theory to non-additive probabilities.

Contribution

It defines finite ergodic components for upper probabilities and proves their equivalence with invariance regimes, eigenvalue multiplicity, and classical ergodic properties.

Findings

Finite ergodic components characterized by capacity regimes.

Eigenvalue 1 of Koopman operator has finite multiplicity in FEC.

Law of large numbers and asymptotic independence linked to FEC.

Abstract

Under the notion of ergodicity of upper probability in the sense of Feng and Zhao (2021) that any invariant set either has capacity $0$ or its complement has capacity 0, we introduce the definition of finite ergodic components (FEC). We prove an invariant upper probability has FEC if and only if it is in the regime that any invariant set has either capacity $0$ or capacity $1$, proposed by Cerreia-Vioglio, Maccheroni, and Marinacci (2016). Furthermore, this is also equivalent to that the eigenvalue $1$ of the Koopman operator is of finite multiplicity, while in the ergodic upper probability regime, as in the classical ergodic probability case, the eigenvalue $1$ of the Koopman operator is simple. Additionally, we obtain the equivalence of the law of large numbers with multiple values, the asymptotic independence and the FEC. Furthermore, we apply these to obtain the corresponding results for non-invariant probabilities.