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Marinacci, Massimo

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

J. Econ. Theory 84, No. 2, 145-195 (1999).
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.

Cited in 3 Reviews
Cited in 77 Documents

MSC:

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

Keywords:

limit laws; non-additive probabilities
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\textit{M. Marinacci}, J. Econ. Theory 84, No. 2, 145--195 (1999; Zbl 0921.90005)
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References:

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