This paper claims:
If we have two random variables ξ1 and ξ2, then we can form their mixture if we take ξ1 with some probability w and ξ2 with the remaining probability 1 − w. The probability density function (pdf) ρ(x) of the mixture is a convex combination of the pdfs of the original variables: ρ(x) = w · ρ1(x) + (1 − w) · ρ2(x). A natural question is: can we use other functions f (ρ1, ρ2) to combine the pdfs, i.e., to produce a new pdf ρ(x) = f (ρ1(x), ρ2(x))? In this paper, we prove that the only combination operations that always lead to a pdf are the operations f (ρ1, ρ2) = w · ρ1 + (1 − w) · ρ2
However, from any standard text book on statistics,
The distribution of the sum of the values of two or more underlying random variables is given by the convolution operator.
Is the claim made in the paper correct?
