The topological space of Schur-concave copulas is homeomorphic to the Hilbert cube. (English) Zbl 07880011
Schur-concave ccopulas were introduced in [F. Durante and C. Sempi, Int. Math. J. 3, No. 9, 893–905 (2003; Zbl 1231.60014)], see the book [A. W. Marshall and I. Olkin, Inequalities: theory of majorization and its applications. New York etc.: Academic Press (1979; Zbl 0437.26007)] for the definition of Schur-concavity. It is known that the subset \(C_{SC}\) of Schur-concave copulas is compact in the topology of uniform convergence. The main result of this paper is that \(C_{SC}\) is homeomorphic to the Hilbert cube, and, as a consequence, has the fixed point property. The methods used in the proof are from infinite-dimensional topology.
Reviewer: Carlo Sempi (Lecce)
MSC:
| 54C35 | Function spaces in general topology |
| 57N20 | Topology of infinite-dimensional manifolds |
| 54E70 | Probabilistic metric spaces |
| 60E05 | Probability distributions: general theory |
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