Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach

Version 1 : Received: 12 October 2019 / Approved: 13 October 2019 / Online: 13 October 2019 (16:45:08 CEST)

A peer-reviewed article of this Preprint also exists.

Vitali, S.; Budimir, I.; Runfola, C.; Castellani, G. The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach. Mathematics 2019, 7, 1145. Vitali, S.; Budimir, I.; Runfola, C.; Castellani, G. The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach. Mathematics 2019, 7, 1145.

Journal reference: Mathematics 2019, 7, 1145
DOI: 10.3390/math7121145

## Abstract

The central limit theorem (CLT) and its generalization to stable distributions have been widely described in literature. However, many variations of the theorem have been defined and often their applicability in practical situations is not straightforward. In particular, the applicability of the CLT is essential for a derivation of heterogeneous ensemble of Brownian particles (HEBP). Here, we analyze the role of the CLT within the HEBP approach in more detail and derive the conditions under which the existing theorems are valid.

## Keywords

central limit theorem; anomalous diffusion; stable distribution; fractional calculus; power law

## Subject

MATHEMATICS & COMPUTER SCIENCE, Probability and Statistics

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