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Limit theorems for fuzzy random variables. (English) Zbl 0605.60038

The authors continue to study fuzzy random variables as defined by the last two authors [J. Math. Anal. Appl. 114, 409-422 (1986; Zbl 0592.60004)]. First they prove a strong law of large numbers for fuzzy random variables. The crucial point is that the space of normal, upper semicontinuous, compactly supported fuzzy sets on ${ℝ}^{n}$ is a separable metric space when equipped with a suitable metric.

The second main result is a central limit theorem. The proof is based on an embedding theorem, which says that under additional assumptions the above mentioned space of fuzzy sets can be embedded isometrically into a space of continuous functions. The embedding theorem has already appeared in a previous paper by the last two authors [Ann. Probab. 13, 1373-1379 (1985; Zbl 0583.60011)] though not referred to in the text.

H. Kwakernaak [Inf. Sci. 15, 1-29 (1978; Zbl 0438.60004)] introduced a different notion of a fuzzy random variable. The strong law of large numbers for these fuzzy random variables is given by R. Kruse [ibid. 28, 233-241 (1982; Zbl 0571.60039)].

Reviewer: O.Kaleva

MSC:
 60F15 Strong limit theorems 60F05 Central limit and other weak theorems 03E72 Fuzzy set theory