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

Klement, E. P.; Puri, M. L.; Ralescu, D. A.

Proc. R. Soc. Lond., Ser. A 407, 171-182 (1986).

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 \({\mathbb{R}}^ n\) 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	Theory of fuzzy sets, etc.

Keywords:

fuzzy random variables; strong law of large numbers; central limit theorem

Citations:

Zbl 0592.60004; Zbl 0583.60011; Zbl 0438.60004; Zbl 0571.60039