Donsker’s fuzzy invariance principle under the Lindeberg condition

Abstract

Abstract We prove an analogue of the Donsker theorem under the Lindeberg condition in a fuzzy setting. Specifically, we consider a certain triangular system of d-dimensional fuzzy random variables { X n , i ∗ } , $\begin{array}{} \{X_{n,i}^*\}, \end{array}$ n ∈ ℕ and i = 1, 2, …, kn , which take as their values fuzzy vectors of compact and convex α-cuts. We show that an appropriately normalized and interpolated sequence of partial sums of the system may be associated with a time-continuous process defined on the unit interval t ∈ [0, 1] which, under the assumption of the Lindeberg condition, tends in distribution to a standard Brownian motion in the space of support functions.