The central limit theorem for capacities. (English) Zbl 1170.60016

Summary: In investigations where the parameter of interest is the mean or expectation of some random variable and the underlying probability measure (distribution) is unknown, one usually appeals to the central limit theorem, provided it holds. In this article, the central limit theorem and the weak law of large numbers for capacities are presented. Capacities are non-additive probability measures which provide alternative and plausible measures of likelihood or uncertainty when the assumption of additivity is suspect. Some examples of practical problems in game theory, economics and finance that can be solved at least partially, by the central limit theorem for capacities, are presented.


60F05 Central limit and other weak theorems
Full Text: DOI


[2] Augustin, T., Optimal decisions under complex uncertainty — basic notions and a general algorithm for data-based decision making with partial prior knowledge described by interval probability, Gesellschaft für Angewandte Mathematik und Mechanik, 84, 678-687 (2004) · Zbl 1056.62009
[3] Choquet, G., The theory of capacities, Ann. Inst. Fourier, 5, 131-295 (1954) · Zbl 0064.35101
[4] Chung, K. L., A Course in Probability Theory (1974), Academic Press: Academic Press New York · Zbl 0159.45701
[5] Doob, J., Classical potential theory and its probabilistic counterpart (1984), Springer: Springer New York · Zbl 0549.31001
[6] Hogg, R. V.; McKean, J. W.; Craig, A. T., Introduction to Mathematical Statistics (2005), Pearson Prentice Hall: Pearson Prentice Hall New Jersey
[7] Huber, P. J.; Strassen, V., Minimax tests and Neyman-Pearson lemma for capacities, Ann. Statist., 1, 2, 251-263 (1973) · Zbl 0259.62008
[8] Huber, P. J.; Strassen, V., Correction to minimax tests and Neyman-Pearson lemma for capacities, Ann. Statist., 2, 1, 223-224 (1974) · Zbl 0269.62020
[9] Macherroni, F.; Marinacci, M., The strong law of large numbers for capacities, Ann. Probab., 33, 3, 1171-1178 (2005) · Zbl 1074.60041
[10] Marinacci, M., Limit laws for non-additive probabilities and their frequentist interpretation, J. Econom. Theory, 84, 145-195 (1999) · Zbl 0921.90005
[11] Ross, S., A first Course in Probability (2006), Pearson Prentice Hall: Pearson Prentice Hall New Jersey · Zbl 1307.60001
[12] Schmeidler, D., Subjective probability and expected utility without additivity, Econometrica, 57, 571-587 (1989) · Zbl 0672.90011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.