Limit theorems for fuzzy-random variables

Authors:
Volker Krätschmer
Affiliations:
Statistik und Ökonometrie im Fachbereich Wirtschaftswissenschaft, Universität des Saarlandes, Bau 31, Postfach 151150, Saarbrücken, Germany
Venue:
Fuzzy Sets and Systems
Year:
2002

Citing 3
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Abstract

This paper deals with limit theorems for fuzzy-valued measurable mappings which provide, as a whole, a foundation of statistical analysis with fuzzy data. A strong law of large numbers, a central limit theorem and a Gliwenko-Cantelli theorem are proved. The results are formulated simultaneously with respect to the Lp-metrics on the fuzzy sample spaces, investigated by Diamond and Kloeden. In particular, these versions of the limit theorems are related to identical, compatible concepts of convergence and measurability in the fuzzy sample spaces. The proofs of the theorems are based heavily on isomorphic isometric embeddings of the fuzzy sample spaces, endowed with Lp-metrics, into respective Lp-spaces, which are Banach spaces of type 2. These embeddings provide the application of convergence results in Banach spaces.