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Finite ergodic components for upper probabilities

Chunrong Feng, Wen Huang, Chunlin Liu, Huaizhong Zhao

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.

Finite ergodic components for upper probabilities

Abstract

Under the notion of ergodicity of upper probability in the sense of Feng and Zhao (2021) that any invariant set either has capacity 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 or capacity , proposed by Cerreia-Vioglio, Maccheroni, and Marinacci (2016). Furthermore, this is also equivalent to that the eigenvalue of the Koopman operator is of finite multiplicity, while in the ergodic upper probability regime, as in the classical ergodic probability case, the eigenvalue 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.

Paper Structure

This paper contains 9 sections, 20 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Capacity and Choquet integral
  4. Invariant skeleton
  5. Convex analysis
  6. Local ergodic decomposition of invariant upper probabilities
  7. Finite ergodic components in terms of multiplicity of the eigenvalue of Koopman operator
  8. Finite ergodic components in terms of independence
  9. Application to non-invariant probabilities

Key Result

Lemma 2.1

Let $(\Omega, \mathcal{F})$ be a measurable space, and $\{P_n\}_{n\in\mathbb{N}}$ be a sequence of probabilities. Suppose that for each $A\in \mathcal{F}$, the limit $\lim _{n \rightarrow \infty} P_n(A)$ exists, denoted by $Q(A)$. Then $Q$ is a probability on $(\Omega,\mathcal{F})$. If we further su

Theorems & Definitions (46)

  • Definition 1.1
  • Remark 1.2
  • Lemma 2.1: Vitali-Hahn-Saks
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Lemma 2.7: Krein-Milman Theorem
  • ...and 36 more