On central limit theorems for IV-events

Abstract

Interval-valued fuzzy sets were introduced in 1970s as an extension of Zadeh's fuzzy sets. For intervalvalued fuzzy events, (IV-events for short) IV-probability theory has been developed. In this paper, we prove central limit theorems for triangular arrays of IV-observables within this theory. We prove the Lindeberg CLT and the Lyapunov CLT, assuming that IV-observables are not necessary identically distributed. We also prove the Feller theorem for null arrays of IV-observables. Furthermore, we present examples of applications of the aforementioned theorems. In particular, we study the convergence in distribution of scaled sums of identically distributed IV-observables.

Interval-based, nonparametric approach for resampling of fuzzy numbers

“…There exist many extensions of fuzzy sets which can also be used for this purpose. For example, interval-valued fuzzy sets (IVFS), introduced independently by four different authors in 1975-see Nowak and Hryniewicz (2018) for references, can be used in situations, when the membership function of a fuzzy set cannot be precisely defined. Another very popular extension, widely known under the name of intuitionistic fuzzy sets (IFS), was introduced in Atanassov (1986) and can be used when we describe imprecision in terms of membership and nonmembership functions.…”

On MV-Algebraic Versions of the Strong Law of Large Numbers

“…The MV-algebraic probability theory was also applied in the Atanassov intuitionistic fuzzy sets and interval-valued fuzzy sets settings (see, e.g., [ 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 ]). The notion of probability for the Atanassov intuitionistic fuzzy sets was introduced by Szmidt and Kacprzyk [ 19 ].…”