[Paper Review] Ergodic Theorems for Lower Probabilities

Simone Cerreia‐Vioglio, Fabio Maccheroni+1 more|arXiv (Cornell University)|Aug 6, 2015

Probability and Statistical Research15 references4 citations

TL;DR

This paper establishes an Ergodic Theorem for lower probabilities, a non-additive generalization of standard probability measures, by introducing four notions of invariance and proving that time averages of bounded measurable functions converge almost surely under the lower probability. The key contribution is a nonadditive version of the Strong Law of Large Numbers, with convergence bounds derived via Choquet integrals under ergodicity.

ABSTRACT

We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a by-product, we provide a version for lower probabilities of the Strong Law of Large Numbers.

Motivation & Objective

• To generalize ergodic theory to lower probabilities, which are non-additive set functions used when standard additive probabilities fail in modeling uncertainty.
• To define and analyze multiple notions of invariance for lower probabilities, showing their equivalence in the additive case but distinct roles in nonadditive settings.
• To establish a nonadditive ergodic theorem where time averages converge ν-a.s. on a set of lower probability one.
• To derive a nonadditive version of the Strong Law of Large Numbers for stationary and ergodic processes under lower probability measures.
• To characterize the limit of time averages using lower and upper Choquet integrals under ergodicity and stronger invariance conditions.

Proposed method

• Introduces four definitions of invariance for lower probabilities, with Definition 1 being the weakest and sufficient for the main ergodic result.
• Uses the Choquet integral to bound the limit of time averages, leveraging the conjugate upper probability and core of the lower probability.
• Applies Kingman’s super-subadditive ergodic theorem in a nonadditive setting to derive convergence under stronger invariance conditions.
• Employs continuity and compactness arguments in the weak* topology on the core of the lower probability to ensure convergence of integrals.
• Establishes convergence of time averages by showing that the set of convergence has lower probability one using properties of convex capacities and continuity at S.
• Uses measurable selection and uniform boundedness to justify interchange of limit and integral via convergence theorems in the context of the core of the lower probability.

Experimental results

Research questions

• RQ1 Can an ergodic theorem be established for lower probabilities, which are non-additive and continuous set functions?
• RQ2 How do different notions of invariance for lower probabilities relate, and which are sufficient to ensure convergence of time averages?
• RQ3 What is the nonadditive analog of the Strong Law of Large Numbers under lower probability measures?
• RQ4 How can the limit of time averages be bounded using Choquet integrals when the lower probability is ergodic?
• RQ5 Under what conditions does the limit of time averages exist ν-a.s. for bounded measurable functions under a lower probability?

Key findings

• The time average of a bounded measurable function converges ν-almost surely to a limit that lies between the lower and upper Choquet integrals of the function under ergodicity.
• For shift-invariant and ergodic lower probabilities, the limit of time averages is almost surely bounded by the Choquet integral of the limit function with respect to the lower and upper probabilities.
• When the underlying space is standard Borel and the lower probability is continuous at S, the limit of time averages is characterized precisely via the core of the lower probability.
• The paper proves a nonadditive version of the Strong Law of Large Numbers: for stationary and ergodic processes, the empirical mean converges to an interval bounded by the lower and upper Choquet integrals of the first random variable.
• Under stronger invariance conditions, the convergence of time averages is guaranteed with full lower probability one, and the limit is measurable with respect to the invariant σ-algebra.
• The limit function is shown to be measurable and the integral of the limit equals the limit of the integrals via uniform boundedness and convergence theorems in the core of the lower probability.