25
More on fuzzy random variables and fuzzy random vectors
For model-based inference methods in time series analysis the concept of fuzzy random variables is useful. Basic definitions and some propositions concerning fuzzy random variables are given in Chapter (A Law of Large Numbers).
For generalized stochastic methods in time series analysis more details on fuzzy random variables are necessary. These will be explained in this section.
25.1 Basics
Definition 25.1:
A fuzzy random vector (also called an n-dimensional fuzzy random variable) is a mapping X from some probability space (Ω,E,
) into the space (
(
n), h∞) with h∞(·,·) the metric defined in (23.4) if X is measurable relative to the σ-field in (n) which is generated by the open sets
with y* ∈ (n) and r > 0.
Remark 25.1:
Using ((n),
