A strong law of large numbers for fuzzy random variables
type: Item
instance of: scholarly article
title: A strong law of large numbers for fuzzy random variables
zbMATH Open document ID: 0551.60035
DOI: 10.1016/0165-0114(84)90033-2
author: Masaaki Miyakoshi
author: Masaru Shimbo
published in: Fuzzy Sets and Systems
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60A99
Mathematics Subject Classification ID: 94D05
zbMATH DE Number: 3878074
zbMATH Keywords: fuzzy random variables
zbMATH Keywords: fuzzy law of large numbers
MaRDI profile type: Publication
full work available at URL: 0165-0114(84)90033-2
OpenAlex ID: W2026578432
cites work: Fuzzy random variables - II. Algorithms and examples for the discrete case
cites work: Convex fuzzy sets
cites work: Fuzzy variables
cites work: A note on the extension principle for fuzzy sets
cites work: Convex fuzzy random variables
cites work: The concept of a linguistic variable and its application to approximate reasoning. I
description: scientific article; zbMATH DE number 3878074