Change of measure technique in characterizations of the gamma and Kummer distributions

Abstract

If X and Y are independent random variables with distributions μ nd ν then U=ψ(X,Y) and V=ϕ(X,Y) are also independent for some transformations ψ and ϕ. Properties of this type are known for many important probability distributions μ and ν. Also related characterization questions have been widely investigated. These are questions of the form: Let X and Y be independent and let U and V be also independent. Are the distributions of X and Y necessarily μ and ν, respectively? Recently two new properties and characterizations of this kind involving the Kummer distribution appeared in the literature. For independent X and Y with gamma and Kummer distributions Koudou and Vallois in [17] observed that U=(1+(X+Y)−1)/(1+X−1) and V=X+Y are also independent, and Hamza and Vallois in [13] observed that U=Y/(1+X) and V=X(1+Y/(1+X)) are independent. In [16] and [17] characterizations related to the first property were proved, while the characterizations in the second setting have been recently given in [29]. These results were not fully satisfactory since in both cases technical assumptions on smoothness properties of densities of X and Y were needed. In [31], the assumption of independence of U and V in the first setting was weakened to constancy of regressions of U and U−1 given V with no density assumptions. However, the additional assumption EX−1<∞ was introduced. In the present paper we provide a complete answer to the characterization question in both settings without any additional technical assumptions regarding smoothness or existence of moments. The approach is, first, via characterizations exploiting some conditions imposed on regressions of U given V, which are weaker than independence, but for which moment assumptions are necessary. Second, using a technique of change of measure we show that the moment assumptions can be avoided.