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

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.

[1]  Jorge Nocedal,et al.  An interior algorithm for nonlinear optimization that combines line search and trust region steps , 2006, Math. Program..

[2]  Alexis Akira Toda.  Weak Limit of the Geometric Sum of Independent But Not Identically Distributed Random Variables , 2011, 1111.1786.

[3]  William J. Reed,et al.  The Double Pareto-Lognormal Distribution—A New Parametric Model for Size Distributions , 2004, WWW 2001.

[4]  H. Teicher.  Identifiability of Finite Mixtures , 1963 .

[5]  O. Barndorff-Nielsen.  Exponentially decreasing distributions for the logarithm of particle size , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[6]  V. Kruglov,et al.  Weak Convergence of Random Sums , 2002 .

[7]  D. Hartmann.  Cross-shore selective sorting processes and grain size distributional shape , 1991 .

[8]  R. E. Carlson,et al.  Monotone Piecewise Cubic Interpolation , 1980 .

[9]  J. Graf,et al.  Lunar soils grain size catalog , 1993 .

[10]  Peter Vincent,et al.  Differentiation of modern beach and coastal dune sands—a logistic regression approach using the parameters of the hyperbolic function , 1986 .

[11]  V. Korolev,et al.  Convergence of Random Sequences with Independent Random Indices II , 1995 .

[12]  E. N. Slyuta,et al.  Physical and mechanical properties of the lunar soil (a review) , 2014, Solar System Research.

[13]  V. Korolev,et al.  On convergence of the distributions of random sequences with independent random indexes to variance–mean mixtures , 2014, 1410.1022.

[14]  E. Zubko,et al.  Formation of Dusty Plasma Clouds at Meteoroid Impact on the Surface of the Moon , 2018, JETP Letters.

[15]  David S. McArthur.  Distinctions between grain-size distributions of accretion and encroachment deposits in an inland dune , 1987 .