The law of large numbers for fuzzy processes and the estimation problem

We study how the law of large numbers can be extended to the case of fuzzy numbers. We work with L-fuzzy sets, where L is a CL_∞ commutative monoid with an operation 1. We define the 1-independence of fuzzy numbers. We find, when 1 is the operation Λ of the lattice, that the arithmetical mean of Λ-independent fuzzy numbers does not converge. We think that this case could be a limit one; we show, for particular cases of the 1-operation, that generally convergence holds. We build estimators for fuzzy models; we particularly consider the “minimum fuzziness estimator.” We study the convergence of these estimators and compare them by way of the fuzzy subsets they induce. We lastly study the case where we have dependent fuzzy numbers and particularly the case of linearly filtered “1-autoindependent” fuzzy processes.