A strong law of large numbers for fuzzy random variables
The concept of fuzzy random variables was first introduced by \textit{H. Kwakernaak} [Inf. Sci. 15, 1-29 (1978; Zbl 0438.60004)]. Three years later \textit{W. E. Stein} and \textit{K. Talati} [Fuzzy Sets Syst. 6, 271-283 (1981; Zbl 0467.60005)] gave another definition of fuzzy random variables and showed a fuzzy law of large numbers. Whether the law of large numbers holds also for Kwakernaak's fuzzy number remains open. The aim of this study is to answer this question. Having discussed some properties of fuzzy numbers, the authors prove a strong law of large numbers: Let \(\{\xi_ n,n\geq 1\}\) be an independent sequence of fuzzy random variables satisfying the following two conditions: (1) \(E\xi_ n=E\xi_ 1\) for any \(n\geq 1\), (2) \(V(\bar X_{{\bar \alpha}}^{(n)})\), \(V(X_{{\bar \alpha}}^{(n)})\leq v_{\alpha}<\infty\) for any \(n\geq 1\) and \(\alpha\in (0,1]\), where \(V(\bar X_{{\bar \alpha}}^{(n)})\) and \(V(X_{{\bar \alpha}}^{(n)})\) stand for variances of random variables \(\bar X_{{\bar \alpha}}^{(n)}\) and \(X_{{\bar \alpha}}^{(n)}\), and \(v_{\alpha}\) stands for a finite nonnegative number. Then \(\lim_{n\to \infty}n^{-1}(\sum^{n}_{k=1}\xi_ k)(\omega)=E\xi_ 1\) with probability 1. Finally, a statistical example is given.
scientific article; zbMATH DE number 3878074