## 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

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### References:

 [1] Choquet, G., Theory of capacities, Ann. Inst. Fourier, 5, 131-295 (1953) · Zbl 0064.35101 [2] de Finetti, B., Teoria della Probabilità (1970), Einaudi: Einaudi Turin [3] Delbaen, F., Convex games and extreme points, J. Math. Anal. Appl., 45, 210-233 (1974) · Zbl 0337.90084 [4] Dempster, A., Upper and lower probabilities induced from a multivalued mapping, Ann. Math. Statist., 38, 325-339 (1967) · Zbl 0168.17501 [5] Dempster, A., A generalization of Bayesian inference, J. Roy. Statist. Soc. Ser. B, 30, 205-247 (1968) · Zbl 0169.21301 [6] Denneberg, D., Non-Additive Measure and Integral (1994), Kluwer: Kluwer Dodrecht · Zbl 0826.28002 [8] Dunford, N.; Schwartz, J. T., Linear Operators, Part 1 (1957), Interscience: Interscience New York [9] Eichberger, J.; Kelsey, D., Uncertainty aversion and preference for randomization, J. Econ. Theory, 71, 370-381 (1996) [10] Ellsberg, D., Risk, ambiguity, and the Savage axioms, Quart. J. Econ., 75, 643-669 (1961) · Zbl 1280.91045 [11] Feynman, R.; Hibbs, A. R., Quantum Mechanics and Path Integrals (1965), McGraw-Hill: McGraw-Hill New York · Zbl 0176.54902 [12] Ghirardato, P., On independence for non-additive measures, with a Fubini theorem, J. Econ. Theory, 73, 261-291 (1997) · Zbl 0934.28012 [13] Gilboa, I., Expected Utility with purely subjective non-additive probabilities, J. Math. Econ., 16, 65-88 (1987) · Zbl 0632.90008 [14] Gilboa, I.; Schmeidler, D., Maxmin expected utility with a non-unique prior, J. Math. Econ., 18, 141-153 (1989) · Zbl 0675.90012 [15] Gilboa, I.; Schmeidler, D., Canonical representation of set functions, Math. Oper. Res., 20, 197-212 (1995) · Zbl 0834.90141 [16] Greco, G., Sur la mesurabilité d’une fonction numérique par rapport à une famille d’ensembles, Rend. Sem. Mat. Univ. Padova, 65, 163-176 (1981) · Zbl 0507.28005 [17] Greco, G., Sulla rappresentazione di funzionali mediante integrali, Rend. Sem. Mat. Univ. Padova, 66, 21-41 (1982) · Zbl 0524.28016 [18] Huber, P. J., The use of Choquet capacities in statistics, Bull. Inst. Internat. Statist., 45, 181-191 (1973) [19] Huber, P. J., Robust Statistical Procedures (1996), SIAM: SIAM Philadelphia [20] Kelley, J. L., Measures on Boolean algebras, Pacific J. Math., 9, 1165-1177 (1959) · Zbl 0087.04801 [21] Kreps, D., Notes on the Theory of Choice (1988), Westview Press: Westview Press Boulder [22] Marinacci, M., Decomposition and representation of coalitional games, Math. Oper. Res., 21, 1000-1015 (1996) · Zbl 0868.90152 [24] Pap, E., Null-Additive Set Functions (1995), Kluwer: Kluwer Dordrecht · Zbl 0856.28001 [25] Savage, L. J., The Foundations of Statistics (1954), Wiley: Wiley New York · Zbl 0121.13603 [27] Schmeidler, D., Integral representation without additivity, Proc. Amer. Math. Soc., 97, 253-261 (1986) · Zbl 0687.28008 [28] Schmeidler, D., Subjective probability and expected utility without additivity, Econometrica, 57, 571-587 (1989) · Zbl 0672.90011 [29] Seidenfeld, T.; Wasserman, L., Dilation for sets of probabilities, Ann. Statist., 21, 1139-1154 (1993) · Zbl 0796.62005 [30] Shafer, G., A Mathematical Theory of Evidence (1976), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0359.62002 [31] Shapley, L. S., Cores of convex games, Internat. J. Game Theory, 1, 11-26 (1971) · Zbl 0222.90054 [32] Vitali, G., Sulla definizione d’integrale delle funzioni di una variabile reale, Ann. Mat. Pura Appl. (IV), 2, 111-121 (1925) · JFM 51.0201.01 [33] Wakker, P., Additive Representation of Preferences (1989), Kluwer: Kluwer Dordrecht [34] Walley, P., Statistical Reasoning with Imprecise Probabilities (1991), Chapman and Hall: Chapman and Hall London · Zbl 0732.62004 [35] Walley, P.; Fine, T. L., Towards a frequentist theory of upper and lower probability, Ann. Statist., 10, 741-761 (1982) · Zbl 0488.62004 [36] Wasserman, L.; Kadane, J., Bayes’ theorem for Choquet capacities, Ann. Statist., 18, 1328-1339 (1990) · Zbl 0736.62026 [37] Zhou, L., Integral representation of continuous comonotonically additive set functions, Trans. Amer. Math. Soc., 350, 1811-1822 (1998) · Zbl 0905.28006
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