A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation

Let $X$ be a fuzzy set--valued random variable (\frv{}), and $\huku{X}$ the family of all fuzzy sets $B$ for which the Hukuhara difference $X\HukuDiff B$ exists $\mathbb{P}$--almost surely. In this paper, we prove that $X$ can be decomposed as $X(\omega)=C\Mink Y(\omega)$ for $\mathbb{P}$--almost every $\omega\in\Omega$, $C$ is the unique deterministic fuzzy set that minimizes $\mathbb{E}[d_2(X,B)^2]$ as $B$ is varying in $\huku{X}$, and $Y$ is a centered \frv{} (i.e. its generalized Steiner point is the origin). This decomposition allows us to characterize all \frv{} translation (i.e. $X(\omega) = M \Mink \indicator{\xi(\omega)}$ for some deterministic fuzzy convex set $M$ and some random element in $\Banach$). In particular, $X$ is an \frv{} translation if and only if the Aumann expectation $\mathbb{E}X$ is equal to $C$ up to a translation. Examples, such as the Gaussian case, are provided.