Fuzzy Set-valued Random Variables Research Articles

Limit Theorems for Negatively Dependent Fuzzy Set-Valued Random Variables

In this paper, we shall discuss negatively dependent fuzzy set-valued random variables. And at last, we shall prove the limit theorems for rowwise negatively dependent fuzzy set-valued random variables in the sense of $d_H^\infty$, which is the extension of (Guan \& Sun, 2014) and (Guan \& Wan, 2016).

Strong Laws of Large Numbers for Fuzzy Set-Valued Random Variables in G<sub>α</sub> Space

In this paper, we shall present the strong laws of large numbers for fuzzy set-valued random variables in the sense of d∞H . The results are based on the result of single-valued random variables obtained by Taylor [1] and set-valued random variables obtained by Li Guan [2].

Strong laws of large numbers for t-norm-based addition of fuzzy set-valued random variables

This paper proposes a convergence theorem of fuzzy set-valued random variables under addition which is given by the general triangular normed extension principles. Applying the result with the known results of the various types for strong laws of large numbers of fuzzy set-valued random variables, we obtain a few interesting results of the strong laws of large numbers for triangular norm-based addition of fuzzy set-valued random variables. The finding of Terán [Strong law of large numbers for t-normed arithmetics, Fuzzy Sets and Systems 159 (2008) 343–360] is a special case of our results.

A decomposition theorem for fuzzy set-valued random variables

In this paper, a decomposition theorem for a (square integrable) fuzzy random variable FRV is proposed. The paper is mainly divided into two parts. In the first part, for any FRV X, we define the Hukuhara set as the family of (deterministic) fuzzy sets C for which the Hukuhara difference X⊖HC exists almost surely; in particular, we prove that such a family is a closed (with respect to different well known metrics) convex subset of the family of all fuzzy sets. In the second part, we prove that any square integrable FRV can be decomposed, up to a random translation, as the sum of a FRV Y and an element C′ chosen uniquely (thanks to a minimization argument) in the Hukuhara set. This decomposition allows us to characterize all fuzzy random translations; in particular, a FRV is a fuzzy random translation if and only if its Aumann expectation equals C′ (given by the above decomposition) up to a deterministic translation. Examples and open problems are also presented.

The Concepts of Tightness for Fuzzy Set Valued Random Variables

In this paper, we introduce several concepts of tightness for a sequence of random variables taking values in the space of normal and upper-semicontinuous fuzzy sets with compact support in <TEX>$R^p$</TEX> and give some characterizations of their concepts. Also, counter-examples for the relationships between the concepts of tightness are given.

A strong law of large numbers of fuzzy set-valued random variables with slowly varying weights

In this paper, we shall present a strong law of large numbers (SLLN) of fuzzy set-valued random variables with slowly varying weights in the sense of the extended Hausdorff metric d∞H, which implies some of results in Fu and Zhang (2007).

Fuzzy set-valued Gaussian processes and Brownian motions

Gaussian processes and Brownian motion are concepts and tools in modelling important uncertain systems in many areas. In view of uncertainty complexity in many real-world problems, we extend these tools to the case where stochastic processes can take on fuzzy sets as values. In this paper, we discuss fuzzy set-valued Gaussian processes based on the results of [S. Li, Y. Ogura, V. Kreinovich, Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Kluwer Academic Publishers, Dordrecht, 2002; S. Li, Y. Ogura, H.T. Nguyen, Gaussian processes and martingales for fuzzy valued variables with continuous parameter, Inform. Sci. 133 (2001) 7–21; S. Li, Y. Ogura, F.N. Proske, M.L. Puri, Central limit theorems for generalized set-valued random variables, J. Math. Anal. Appl. 285 (2003) 250–263; N.N. Lyashenko, On limit theorems for sums of independent compact random subsets in the Euclidean space, J. Soviet Math. 20 (1982) 2187–2196] and [M.L. Puri, D.A. Ralescu, The concept of normality for fuzzy random variables, Ann. Probab. 13 (1985) 1373–1379]. We also introduce the concept of fuzzy set-valued Brownian motion, and then prove several properties of such processes.

Strong laws of large numbers for independent fuzzy set-valued random variables

In this paper, we shall present strong laws of large numbers (SLLN's) for independent (not necessary identically distributed) fuzzy set-valued random variables whose base space is a separable Banach space or an Euclidean space, in the sense of the extended Hausdorff metric d H ∞ . We apply the method to the sequence of independent identically distributed fuzzy set-valued random variables to give a simple proof of SLLN's.

A GENERAL METHOD FOR CONVERGENCE THEOREMS OF FUZZY SET-VALUED RANDOM VARIABLES AND ITS APPLICATIONS TO MARTINGALES AND UNIFORM AMARTS

In this paper, we shall first give a general method for convergence theorem of fuzzy set-valued random variables. By using this "sandwich" method, we give the proofs of convergence theorems for fuzzy set-valued martingales, sub- and supermartingales in the sense of the extended Hausdorff metric H∞. Also we shall state a convergence result about uniform amarts.

퍼지 랜덤 집합에 대한 중심극한정리

이 논문에서는 서로 독립이고 동일한 분포를 갖는 집합치 랜덤 변수의 합에 대한 중심극한정리의 일반화로서, 수준연속인 퍼지 집합치 랜덤 변수의 합에 대한 중심극한정리를 연구하였다. The present paper establishes the improved version of central limit theorem for sums of level-continuous fuzzy set-valued random variables as a generalization of central limit theorem for sums of independent and identically distributed set-valued random variables.

LAWS OF LARGE NUMBERS FOR WEIGHTED SUMS OF FUZZY SET-VALUED RANDOM VARIABLES

In this paper, we shall present weak and strong laws of large numbers (WLLN's and SLLN's) for weighted sums of independent (not necessarily identically distributed) fuzzy set-valued random variables in the sense of the extended Hausdorff metric [Formula: see text], based on the result of set-valued random variable obtained by Taylor and Inoue32,33. This work is a continuation of Li and Ogura20.

SEQUENCES OF RANDOM FUZZY SETS

This paper summarizes some results for fuzzy set-valued random variables introduced by Puri and Ralescu.8 Special attention is devoted to their limit properties. The Radon-Nikodym theorem for fuzzy-valued measures is presented as well.