On Kummer’s distribution of type two and a generalized beta distribution

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

“…Finally, we utilize the results obtained under regression assumptions to obtain the characterization by independence without any smoothness or integrability conditions. Another independence property of the Kummer and gamma distributions has been recently discovered in [13]: if X ∼ K(a, b − a, c) and Y ∼ G(b, c) then…”

“…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].…”

Regression conditions that characterizes free--Poisson and free--Kummer distributions

“…An interesting property noticed in [7] says, that if X ∼ K(a, b, c) and Y ∼ G(a + b, c) are independent, then random variables (1) U := Y 1 + X and V := X (1 + U ) are also independent and U ∼ K(a + b, −b, c), V ∼ G(a, c). We call this the HV property for the sake of its authors names.…”

Kummer and gamma laws through independences on trees—Another parallel with the Matsumoto–Yor property

“…In the present paper we are interested in an independence property discovered recently in [10]. It states that if X and Y are independent random variables with Kummer and Gamma distributions, then U = Y /(1 + X) and V = X 1 + X + Y 1 + X are also independent and have Kummer and Gamma distributions, respectively.…”

“…Moreover, we show that if S is a random vector in (0, ∞) p and for any leaf r of the tree the components of Φr(S) are independent, then one of these components has a Gamma distribution and the remaining p − 1 components have Kummer distributions. Our point of departure is a relatively simple independence property due to [10]. It states that if X and Y are independent random variables having Kummer and Gamma distributions (with suitably related parameters) and T : (0, ∞) 2 → (0, ∞) 2 is the involution defined by T (x, y) = (y/(1 + x), x + xy/(1 + x)), then the random vector T (X, Y ) has also independent components with Kummer and gamma distributions.…”