Conditions of Convergence in Distribution for Random Fuzzy Variables

Fuzziness plays an essential role in the real world. Fuzzy set theory has been developed very fast since it was introduced by Zadeh (1965) [1]. A fuzzy set was characterized with its membership function by Zadeh. The term fuzzy variable was fist introduced by Kaufmann (1975) [2], and then appeared in Zadeh (1978) [3] and Nahmias (1978) [4] as a fuzzy set of real numbers. In order to establish the mathematics of fuzzy set theory, Nahmias (1978) [4] introduced three axioms to define possibility spaces. A fuzzy variable may be defined as a function from a possibility space to the set of real numbers. In order to define a selfdual measure, Liu and Liu (2002) [5] gave the concept of credibility measure. And Liu (2004) [6] presented an axiomatic foundation of credibility theory dealing with fuzzy variables based on credibility measure. Fuzzy variable was generalized by bifuzzzy variable, random fuzzy variable, and so on. Bifuzzy variable was introduced by Liu (2002) [7] as a function from a possibility space to the set of fuzzy variables. And random fuzzy variable was defined by Liu (2002) [8] as a function from a possibility space to the set of random variables. Based on the chance measure and expected value operator in Liu (2002) [8] and Liu and Liu (2003) [9], some mathematical properties of random fuzzy variables were derived by Zhu and Liu (2004) [10] [11]. The concept of chance distribution for random fuzzy variables was introduced and several properties of chance distributions were studied in Zhu and Liu (2004) [10]. For random fuzzy sequences, there are several concepts of convergence, for example, convergence almost surely, convergence in chance, convergence in mean and convergence in distribution. It is useful to deal with the criteria of convergence in distribution for random fuzzy sequences. In the following, we first recall some useful concepts such as possibility spaces, random fuzzy variables and chance distributions. Then we investigate some sufficient and necessary conditions of convergence in distribution for random fuzzy sequences.