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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.

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