Laws of large numbers for non-additive probabilities

Abstract

We apply the concept of exchangeable random variables to the case of non-additive robability distributions exhibiting ncertainty aversion, and in the lass generated bya convex core convex non-additive probabilities, ith a convex core). We are able to rove two versions of the law of arge numbers (de Finetti's heorems). By making use of two efinitions. of independence we rove two versions of the strong law f large numbers. It turns out that e cannot assure the convergence of he sample averages to a constant. e then modal the case there is a true' probability distribution ehind the successive realizations of the uncertain random variable. In this case convergence occurs. This result is important because it renders true the intuition that it is possible 'to learn' the 'true' additive distribution behind an uncertain event if one repeatedly observes it (a sufficiently large number of times). We also provide a conjecture regarding the 'Iearning' (or updating) process above, and prove a partia I result for the case of Dempster-Shafer updating rule and binomial trials.