On the problem of characterizing the distribution of random variables by the distribution of their sum (Q1848034)

The aim of this article is to obtain conditions under which the distribution of the sum of \(n\) independently and identically distributed (iid) real random variables uniquely determines the common distribution of the individual variables. Some examples in the form of two theorems are given where the sum of iid variables does not characterize the common distribution of the individual variables. Two theorems are given where the sum of iid variables uniquely determines the common distribution of the individual variables, where one result assumes the existence of the absolute moments of order \(n\) and the other assumes differentiable symmetric characteristic functions. Examples and two auxiliary results are also given.