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

G.G Hamedani,Mahrokh Najaf

doi:10.18187/pjsor.v18i1.3553

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.

Highlights

To make this very short note self contained, we will copy some parts of our previous work, Hamedani(2013), here.We may in some occasions have asked ourselves if there is any concept between ”uncorrelatedness” and ”independence” of two random variables
It seems that the concept of ”sub-independence” is the one: it is much stronger than uncorrelatedness and much weaker than independence
The notion of sub-independence seems important in the sense that under usual assumptions, Khintchine’s Law of Large Numbers and Lindeberg

Summary

Introduction

To make this very short note self contained, we will copy some parts of our previous work, Hamedani(2013), here. Limit theorems as well as other well-known results in probability and statistics are often based on the distribution of the sums of independent (and often identically distributed) random variables rather than the joint distribution of the summands. ˆ height (Y ) is modeled by the sum of a cyclic function of random delay D, g (D), and a residual term ε They found that these two components are at least uncorrelated but not independent and used sub-independence to compute the distribution of the return value. The drawback of the concept of sub-independence in comparison with that of independence has been that the former does not have an equivalent definition in the sense of (1.3) , which some believe, to be the natural definition of independence We found such a definition which is stated below.

Consider the events

Hc are obtained by shifting each point in