zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $\{X_k\}_{k\ge 1}$ are i.i.d. random variables relative to a totally monotone and continuous capacity $v$, then $$v\left(\Bigl\{\int X_1dv\le\lim \inf_n{1 \over n} \sum^n_{k=1} X_k\le\lim \sup_n {1\over n}\sum^n_{k=1} X_k\le-\int-X_1 dv\Bigr\} \right)=1.$$ 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:
91B06Decision theory
60F05Central limit and other weak theorems
WorldCat.org
Full Text: DOI
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)
[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) · 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) · Zbl 0128.34803
[9] Eichberger, J.; Kelsey, D.: Uncertainty aversion and preference for randomization. J. econ. Theory 71, 370-381 (1996) · Zbl 0864.90007
[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) · 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) · Zbl 0859.62003
[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)
[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) · Zbl 0856.28001
[25] Savage, L. J.: The foundations of statistics. (1954) · Zbl 0055.12604
[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) · 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) · Zbl 51.0201.01
[33] Wakker, P.: Additive representation of preferences. (1989) · Zbl 0668.90001
[34] Walley, P.: Statistical reasoning with imprecise probabilities. (1991) · 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