Modeling Particle Size Distribution in Lunar Regolith via a Central Limit Theorem for Random Sums

• Published in: Mathematics (Basel) Vol. 8; no. 9; p. 1409
• Main Authors: Gorshenin, Andrey, Korolev, Victor, Zeifman, Alexander
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2020
• Subjects: Central limit theorem; Cluster analysis; Distribution functions; Empirical analysis; Food science; Grouped observations; Lunar surface; Mathematical models; Normal distribution; Particle size; Particle size distribution; random index; random Lindeberg condition; Random numbers; random sequence; random sum; Random variables; Regolith; Sums; Theorems; transfer theorem
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math8091409

A version of the central limit theorem is proved for sums with a random number of independent and not necessarily identically distributed random variables in the double array limit scheme. It is demonstrated that arbitrary normal mixtures appear as the limit distribution. This result is used to substantiate the log-normal finite mixture approximations for the particle size distributions of the lunar regolith. This model is used as the theoretical background of the two different statistical procedures for processing real data based on bootstrap and minimum χ2 estimates. It is shown that the cluster analysis of the parameters of the proposed models can be a promising tool for revealing the structure of such real data, taking into account the physico-chemical interpretation of the results. Similar methods can be successfully used for solving problems from other subject fields with grouped observations, and only some characteristic points of the empirical distribution function are given.