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A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation
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A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation
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A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation
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Paper
A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation
2011
Overview
Let \\(X\\) be a fuzzy set--valued random variable (\\frv{}), and \\(\\huku{X}\\) the family of all fuzzy sets \\(B\\) for which the Hukuhara difference \\(X\\HukuDiff B\\) exists \\(\\mathbb{P}\\)--almost surely. In this paper, we prove that \\(X\\) can be decomposed as \\(X(\\omega)=C\\Mink Y(\\omega)\\) for \\(\\mathbb{P}\\)--almost every \\(\\omega\\in\\Omega\\), \\(C\\) is the unique deterministic fuzzy set that minimizes \\(\\mathbb{E}[d_2(X,B)^2]\\) as \\(B\\) is varying in \\(\\huku{X}\\), and \\(Y\\) is a centered \\frv{} (i.e. its generalized Steiner point is the origin). This decomposition allows us to characterize all \\frv{} translation (i.e. \\(X(\\omega) = M \\Mink \\indicator{\\xi(\\omega)}\\) for some deterministic fuzzy convex set \\(M\\) and some random element in \\(\\Banach\\)). In particular, \\(X\\) is an \\frv{} translation if and only if the Aumann expectation \\(\\mathbb{E}X\\) is equal to \\(C\\) up to a translation. Examples, such as the Gaussian case, are provided.
Publisher
Cornell University Library, arXiv.org
Subject
