The Remarks on Seven New Publications Based on Sub-Independence Concept

Abstract

Chesneau and Palacios considered the infinite decomposability of the Geometric (Chesneau and Palacios(2021b), (paper 1)), and Gamma, Laplace and n-Laplace (Chesneau and Palacios(2021a), (paper 2)) of two (as well as n) independent random variables. They obtained, very nicely, certain important results on the decomposability concept. Also, George Yanev published a paper entitled” Exponential and Hyper exponential Distributions: Some Characterizations” (Yanev G.(2020), (paper 3)) and reported a paper entitled ”On Arnold-Villasenor Conjectures for Characterizing Exponential Distribution Based on Sample of Size Three” (Yanev(2020), (paper 4)). In both papers, George Yanev considered the distribution of the sum or a linear combination of the independent random variables. Yanev obtained certain nice results in these two papers under the assumption of independence of the summands. Roozegar and Bazyani published a paper entitled ”Exact Distribution of Random Weighted Convolution of Some Beta Distributions Through an Integral Transform” (Roozegar and Bazyari(2017), (paper 5)), in which they considered the exact distribution of the weighted average of n independent beta random variables and provided a new integral transformation with some of its mathematical properties. Ahmad et al.(2021) considered ”Compound Negative Binomial Distribution as the Sum of Independent Laplace Variates” (paper 6) and discussed infinite divisibility of the underlining distribution. Furthermore, Marques et al.(2015) considered the distribution of the linear combinations of independent Gumbel random variables and obtain, very nicely, certain important results (paper 7). In this short note, we like to show that the very strong assumption of ”independence” can be replaced with a much weaker assumption of ”sub-independence” in all aforementioned papers. This short paper may be helpful to other investigators dealing with the random variables which are not necessary independent, but could be sub-independent.