Constraints on belief functions imposed by fuzzy random variables

Dempster´s upper and lower probabilities occur when dealing with multivalued mappings and qualify as Shafer´s belief and plausibility measures. In this paper we generalize the notion of upper and lower probabilities while maintaining their properties as belief and plausibility measures, and subsequently apply this concept on Kwakernaak´s fuzzy random variables. It is shown how fuzzy random variables induce belief and plausibility functions and it is pointed out what relations exist between fuzzy probabilities and degrees of belief and plausibility. Finally we focus on empirical fuzzy random variables and investigate their asymptotic behavior in terms of belief and plausibility constraints and gain a framework for nonparametric statistical reasoning on the basis of uncertain evidences