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

• Published in: Journal of mathematical analysis and applications 2018-02, Vol.458 (2), p.967-979
• Main Authors: Piliszek, Agnieszka, Wesołowski, Jacek
• Format: Article
• Language: English
• Subjects: Change of measure; Characterizations of probability distributions; Constancy of regression; Gamma distribution; Kummer distribution; Matsumoto–Yor property
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2017.10.011

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