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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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[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 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 Dodrecht · Zbl 0826.28002
[7] J. Dow, S. R. C. Werlang, Laws of large numbers for non-additive probabilities, 1994
[8] Dunford, N.; Schwartz, J. T., Linear Operators, Part 1, (1957), 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 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 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 Boulder
[22] Marinacci, M., Decomposition and representation of coalitional games, Math. Oper. Res., 21, 1000-1015, (1996) · Zbl 0868.90152
[23] M. Marinacci, Vitali’s early contribution to non-additive integration, Riv. Mat. Sci. Econom. Social.
[24] Pap, E., Null-Additive Set Functions, (1995), Kluwer Dordrecht · Zbl 0856.28001
[25] Savage, L. J., The Foundations of Statistics, (1954), Wiley New York · Zbl 0121.13603
[26] D. Schmeidler, Subjective probability without additivity, Foerder Institute for Economic Research, Tel Aviv University, 1982 · Zbl 0672.90011
[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 · 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 Dordrecht
[34] Walley, P., Statistical Reasoning with Imprecise Probabilities, (1991), 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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