The central limit theorems for fuzzy random variables (Q1969688)

The author finds assumptions on fuzzy random numbers (frn's) which allow to trace central limit theorems (CLT's) for them back to CLT's for classical random variables. Some more detailed: By assumptions on continuity and closedness, a frn \(\widetilde X\) is uniquely described by its \(\alpha\)-cut \([\widetilde X^L_\alpha, \widetilde X^U_\alpha]\), i.e. by the two families \(\{\widetilde X^L_\alpha\}_{\alpha\in [0,1]}\), \(\{\widetilde X^U_\alpha \}_{ \alpha \in[0,1]}\) of classical random variables. The author defines a so-called fuzzy distribution function of \(\widetilde X\), especially the fuzzy expectation \(\widetilde E(\widetilde X)\) by \((\widetilde E(\widetilde X))_\alpha =[E \widetilde X^L_\alpha,E\widetilde X^U_\alpha]\), and calls \(\widetilde X\) normally distributed if \(\widetilde X^L_\alpha\), \(\widetilde X^U_\alpha\) are normally distributed for all \(\alpha\in [0,1]\). The paper presents some CLT's for so-called \(H\)-frn's where \(\widetilde X\) is called \(H\)-frn if the Hukuhara-difference \(\widetilde X(w)\ominus \widetilde E(\widetilde X)\) exists for all \(w\in\Omega\).