Distributions of random sets and random selections. (English) Zbl 0545.60018
A selection of a random set in the complete separable metric space X is defined as a random point belonging to the random set. It is proved that a necessary and sufficient condition for a probability distribution \(\rho\) on X to be a selection of some random set having given probability measure is that, for all closed \(C\subset X\), \(\rho\) (C) is bounded above by the probability of the random set having non-empty intersection with C. The inequalities are analogous to those required for the existence of matchings in the so-called marriage problem. The family of distributions which may arise as selections of a random set measure is convex and compact according to the Prohorov metric, and depends continuously upon the random set measure.
Reviewer: P.J.Dichomides
MSC:
| 60D05 | Geometric probability and stochastic geometry |
References:
| [1] | Billingsley, P., Convergence of Probability Measures (1968), New York: Wiley, New York · Zbl 0172.21201 |
| [2] | Bollobás, B.; Varopoulos, N. Th., Representation of systems of measurable sets, Math. Proc. Cambridge Phil. Soc., 78, 323-325 (1974) · Zbl 0304.28001 · doi:10.1017/S0305004100051756 |
| [3] | Grenander, U., Pattern Analysis—Lectures in Pattern Theory—I (1978), New York: Springer-Verlag, New York · Zbl 0428.68098 |
| [4] | Halmos, P. R.; Vaughan, H. E., The marriage problem, Am. J. Math., 72, 214-215 (1950) · Zbl 0034.29601 · doi:10.2307/2372148 |
| [5] | Hart, S.; Kohlberg, E., Equally distributed correspondences, J. Math. Econ., 1, 167-174 (1974) · Zbl 0288.28013 · doi:10.1016/0304-4068(74)90007-X |
| [6] | Hart, S.; Hildenbrand, W.; Kohlberg, E., On equilibrium allocations as distributions on the commodity space, J. Math. Econ., 1, 159-166 (1974) · Zbl 0292.90014 · doi:10.1016/0304-4068(74)90006-8 |
| [7] | Hildenbrand, W., Core and Equilibria of a Large Economy (1974), Princeton: Princeton University Press, Princeton · Zbl 0351.90012 |
| [8] | Kendall, D. G.; Harding, E. F.; Kendall, D. G., Foundation of a theory of random sets, Stochastic Teometry, 322-376 (1974), London: Wiley, London · Zbl 0275.60068 |
| [9] | Matheron, G., Random Sets and Integral Geometry (1975), London: Wiley, London · Zbl 0321.60009 |
| [10] | Salinetti, G.; Wets, R. J.-B., On the convergence in distribution of measurable multifunctions, normal integrands, processes and stochastic infima (1982), Laxenburg, Austria: International Institute for Applied Systems Analysis, Laxenburg, Austria |
| [11] | Wagner, W., Survey of measurable selection theorems, SIAM J. Control Optim., 15, 859-901 (1977) · Zbl 0407.28006 · doi:10.1137/0315056 |
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