Limit laws for non-additive probabilities and their frequentist interpretation

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 {X-k}(k greater than or equal to 1) are i.i.d. random variables relative to a totally monotone and continuous capacity v, then v({integral X-1 dv less than or equal to lim inf(n) 1/n (k=1)Sigma(n) X-k less than or equal to lim sup(n) 1/n (k=1)Sigma(n) X-k less than or equal to - integral - X-1 dv}) = 1. Since in the additive case integral X-1 dv = - integral - X-1 dv, 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 Literature Classification Numbers: C60, D81. (C) 1999 Academic Press.