On Independence for Capacities to Fit Ellsberg's Model with a Weak Law of Large Numbers

TL;DR

This paper develops new notions of independence under capacities to model Ellsberg's ambiguity, establishing a weak law of large numbers that supports the model's robustness under mean and variance uncertainty.

Contribution

It introduces Fubini and Exponential independence notions under capacities and explores their relations, providing a weak law of large numbers for Ellsberg's model.

Findings

Weak law of large numbers holds under Exponential independence

Ellsberg's model is robust with mean and variance uncertainty

Simulations confirm theoretical results

Abstract

This paper introduces new notions of Fubini independence and Exponential independence of random variables under capacities to fit Ellsberg's model, and finds out the relations between Fubini independence, Exponential independence, MacCheroni and Marinacci's independence and Peng's independence. As an application, we give a weak law of large numbers for capacities under Exponential independence. Simulations show that Ellsberg's model enjoy the weak law of large numbers when there is mean uncertainty with or without variance uncertainty.