## Limit laws for non-additive probabilities and their frequentist interpretation.(English)Zbl 0921.90005

Summary: 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 element of a sequence $$\{X_k\}_{k\geq 1}$$ are i.i.d. random variables relative to a totally monotone and continuous capacity $$v$$, then $v\left(\Bigl\{\int X_1dv\leq\lim \inf_n{1 \over n} \sum^n_{k=1} X_k\leq\lim \sup_n {1\over n}\sum^n_{k=1} X_k\leq-\int-X_1 dv\Bigr\} \right)=1.$ Since in the additive case $$\int X_1dv=-\int-X_1dv$$, 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. $$\copyright$$ Academic Press.

### MSC:

 91B06 Decision theory 60F05 Central limit and other weak theorems