A Proof of Donsker’s Invariance Principle Based on Support Functions of Fuzzy Random Vectors

Abstract

In this note — starting from d-dimensional (with d > 1) fuzzy vectors — we prove Donsker’s classical invariance principle. We consider a fuzzy random walk [Formula: see text], where [Formula: see text] is a sequence of mutually independent and identically distributed d-dimensional fuzzy random variables whose α-cuts are assumed to be compact and convex. Our reasoning and technique are based on the well known conjugacy correspondence between convex sets and support functions, which allows for the association of an appropriately normalized and interpolated time-continuous fuzzy random process with a real valued random process in the space of support functions. We show that each member of the associated family of dual sequences tends in distribution to a standard Brownian motion.