Limit Laws for Non-additive Probabilities and Their Frequentist Interpretation

Abstract In this paper we prove several limit laws for non-additive probabilities. In particular, we prove that, under a multiplicative notion of independence and a regularity condition, if the elements of a sequence {Xk}k⩾1are i.i.d. random variables relative to a totally monotone and continuous capacityν, then ν ∫ X 1 dν⩽lim inf; n 1 n ∑ k=1 n X k ⩽lim sup; n 1 n ∑ k=1 n X k ⩽−∫−X 1 dν: =1. Since in the additive case ∫ X1 dν=−∫−X1 dν, this is an extension of the classic Kolmogorov's Strong Law of Large Numbers to the non-additive case. We argue that this result suggests a frequentist perspective on non-additive probabilities.Journal of Economic LiteratureClassification Numbers: C60, D81.