Convergence of fuzzy random walks to a standard Brownian motion

TL;DR

This paper proves that fuzzy random walks in multiple dimensions converge to a standard Brownian motion, extending classical invariance principles to fuzzy vector spaces using support functions.

Contribution

It establishes a Donsker-type invariance principle for fuzzy random walks in higher dimensions, utilizing support functions and convex set conjugacy.

Findings

Fuzzy random walks converge to Brownian motion in distribution.

Support functions enable the analysis of fuzzy processes.

The approach generalizes classical invariance principles to fuzzy 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 ${S^*_n}=X^*_1+\cdots+X^*_n,$ where $\{X^*_i\}_1^{\infty}$ 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.