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Statistical Methods for Fuzzy Data by Reinhard Viertl

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9.3 Sequences of fuzzy random variables

Sequences of observations are basic for statistical inference. Moreover the law of large numbers makes sure that taking more observations is reasonable. Therefore it is necessary to consider sequences of fuzzy observations. Let (Xn*)nN be a sequence of fuzzy random variables defined on a complete probability space (Ω, , P), such that for every ω ∈ Ω the sequence (Xn*(ω))nN is cut-wise bounded.

In order to prove a generalized law of large numbers for fuzzy valued random variables, independence of fuzzy random variables has to be defined.

Definition 9.2:

Suppose that (Ω, , P) is a complete probability space and X*; Ω → and Y* : Ω → are fuzzy random variables. Then X* and Y* are said to be independent if for arbitrary Borel-sets B1 and B2 the following equality holds for every δ ∈ (0; 1]:

Unnumbered Display Equation

Proposition 9.1:

Let (Ω, , P) be a complete probability space ...

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