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Donsker’s fuzzy invariance principle under the Lindeberg condition

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Published/Copyright: April 14, 2021
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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 {Xn,i},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.

MSC 2010: 60A86; 60F17; 60J65
  1. (Communicated by Gejza Wimmer )

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Received: 2020-02-10
Accepted: 2020-05-26
Published Online: 2021-04-14
Published in Print: 2021-04-27

© 2021 Mathematical Institute Slovak Academy of Sciences

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