Convergence of fuzzy random walks to a standard Brownian motion

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 $S_n^{\ast }=X_1^{\ast }+\cdots +X_n^{\ast },$ where $\{X_i^{\ast }{\}}_1^{\infty }$ is a sequence of mutually independent and identically distributed $d$-dimensional fuzzy random variables whose $\alpha$-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.

A little changed version under the title: "A proof of Donsker's invariance principle bases on support functions of fuzzy random vectors" will appear in International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems