A Strong Law of Large Numbers for Random Sets in Fuzzy Banach Space

Abstract

The main purpose of this paper is to consider the strong law of large numbers for random sets in fuzzy metric space. Since many years ago, limited theorems have been expressed and proved for fuzzy random variables, but despite the uncertainty in fuzzy discussions, the nonfuzzy metric space has been used. Given that the fuzzy random variable is defined on the basis of random sets, in this paper, we generalize the strong law of large numbers for random sets in the fuzzy metric space. The embedded theorem for compact convex sets in the fuzzy normed space is the most important tool to prove this generalization. Also, as a result and by application, we use the strong law of large numbers for random sets in the fuzzy metric space for the bootstrap mean.

A Limit Theorem for Weighted Sums of Random Sets in Fuzzy Metric Space

“…Gregori and Morillas in [10] introduced examples of fuzzy metrics and its applications. In 2020, Ghasemi et al in [11] discussed the laws of large numbers for fuzzy random variables in the sense of fuzzy metric, but they did not gave the definition of fuzzy metric for sets or fuzzy sets in detail. In 2021, Guan etc.…”