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Representation theorems for extremal distributions. (English) Zbl 0546.60018

In this paper, the authors give a new proof of a result of G. C. Taylor [Scand. Actuarial J. 1977, 94-105 (1977; Zbl 0369.62111)] and the second author [Insur. Math. Econ. 1, 109-130 (1982; Zbl 0488.49030)] on the representation of extremal distributions and generalize it as follows:
If E is a compact and metrizable topological space, \(f_ 0,f_ 1,...,f_ n\in C(E)\) and \(S_ 1(z)\neq\emptyset \), then there exists a probability distribution \(\nu\) on (E,\({\mathcal B})\) that is supported by at most \(n+1\) points such that \(\nu\in S_ 1(z)\) and \(\int f_ 0d\nu =\sup\{\int f_ 0d\mu\); \(\mu\in S_ 1(z)\}\) where \(z=(z_ 1,...,z_ n)\in R^ n\), \({\mathcal N}^+_ 1\) is an arbitrary subset of \({\mathcal M}^+_ 1\) (the set of all probability measures on (E,\({\mathcal B}))\) that contains all the probability distributions supported by at most \(n+1\) points of E, and \(S_ 1(z)=\{\mu\in {\mathcal N}^+_ 1;\int f_ id\mu =z_ i\), \(i=1,...,n\}.\)
The authors also consider the case in which E is a locally compact, \(\sigma\) -compact and metrizable topological space.
Reviewer: X.Yu

MSC:

60E05 Probability distributions: general theory
46A99 Topological linear spaces and related structures
60A10 Probabilistic measure theory
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[1] Choquet, G., Lecture on Analysis (1976), Benjamin: Benjamin Reading, MA, Third printing
[2] De Vylder, F., Best upper bounds for integrals with respect to measures allowed to vary under canonical and integral constraints, Insurance: Mathematics Economics, 1, 2, 109-130 (1982) · Zbl 0488.49030
[3] Heilmann, W. R., Improved methods for calculating and estimating maximal stop-loss premiums, Blätter der Deutschen Gessellschaft für Versicherungsmathematiker, 29-41 (1980) · Zbl 0485.62114
[4] Taylor, G. C., Upper bounds on stop-loss premiums under constraints on claim size distribution, Scandinavian Actuarial Journal, 94-105 (1977) · Zbl 0369.62111
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