Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables

By Shoumei Li, Y. Ogura, Vladik Kreinovich. 2002

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Imprecise probabilities in engineering analyses (2013, unknown): “random processes were examined to reveal properties of limit theorems and martingales associated with fuzzy randomness [224, 225]; see also [226, 227] and for a survey [228].”
Asymptotically Efficient Estimation of Models Defined by Convex Moment Inequalities (2014, background): “10 in Li et al. (2002), (162), (163) and (172) imply: lim sup n→∞ E[L( √ ndH(Θ̂n,Θ0(P0)))]− E[L(‖G0‖∞)]| = lim sup n→∞ |E[L(‖Gn‖∞)]− E[L(‖G0‖∞)]| = 0 , (173) which together with result (160) establishes the claim of the Theorem.”
Asymptotic Properties for a Class of Partially Identified Models (2006, unknown): “…level α ∈ (0, 1) , we propose the following criterion: Reject H0 if √ nH µ 1 n nL i=1 coFi,Ψ0 ¶ > cα, Do not Reject H0 if √ nH µ 1 n nL i=1 coFi,Ψ0 ¶ ≤ cα, where cα is chosen so that (3.1) Pr n kzkC(S) > cα o = α. Observe that the test statistic uses the Hausdorff distance between Ψ0…”
Is there a need for fuzzy logic? (2008, background): “There is an extensive literature on fuzzy-set-valued random variables [52,62,25,46,55,9,51].”