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Central limit theorem for Banach space valued fuzzy random variables. (English) Zbl 0990.60020
From the authors’ introduction: The central limit theorem (CLT), given here, is a generalization of the latter authors’ result, because no Lipschitz condition is needed. In order to then face the non-separability of involved metric spaces of fuzzy sets, our main approach is to identify linear isometrically each fuzzy random variable with convex and compact \(\alpha\)-level sets with an empirical process in \(l^\infty(T)\) to solve a Donsker problem [see A. van der Vaart and J. A. Wellner, “Weak convergence and empirical processes” (1996; Zbl 0862.60002)].
W. Weil [Z. Wahrscheinlichkeitstheorie Verw. Geb. 60, 203-208 (1982; Zbl 0481.60018)] proved a CLT for \(R^p\)-valued compact random sets (i.e. random sets whose values are compact subsets of \(R^p\)). Later on these results were extended by E. Gine, M. G. Hahn and J. Zinn [in: Probability in Banach spaces IV. Lect. Notes. Math. 990, 112-135 (1983; Zbl 0521.60022)] and M. L. Puri and D. A. Ralescu [Math. Proc. Camb. Philos. Soc. 97, 151-158 (1985; Zbl 0559.60007)] to the case of separable Banach spaces. Our result is a natural extension of all previous CLT’s for random sets, because no Lipschitz condition is imposed.

MSC:
60F05 Central limit and other weak theorems
46B09 Probabilistic methods in Banach space theory
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