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Statistical Methods for Fuzzy Data by Reinhard Viertl

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9.4 Law of large numbers for fuzzy random variables

In this section we consider sequences of pairwise independent, identically distributed fuzzy random variables X1*, X2*, X3*,…. For such sequences fuzzy relative frequencies hn* (B, ω) are considered which are generated by the following family of sets:

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and

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For every ω ∈ Ω and every Borel-set B the fuzzy relative frequencies hn*(B, ω) are defined as the convex hull of the family

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For the fuzzy relative frequencies the following lemma holds true.

Lemma 9.3:

Suppose that (Ω, , P) is a complete probability space and X1*, X2*, X3*, … are pairwise independent, identically distributed fuzzy random variables, and B is a Borel-set. Let for every δ ∈ (0; 1] be and as in Section 9.2. Then for every δ ∈ (0; 1] there exists an event N ∈ with P(N) = ...

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