Independence properties of the Matsumoto–Yor type

Abstract

We define Letac–Wesolowski–Matsumoto–Yor (LWMY) functions as decreasing functions from (0, ∞) onto (0, ∞) with the following property: there exist independent, positive random variables X and Y such that the variables f(X + Y) and f(X) − f(X + Y) are independent. We prove that, under additional assumptions, there are essentially four such functions. The first one is f(x) = 1/x. In this case, referred to in the literature as the Matsumoto–Yor property, the law of X is generalized inverse Gaussian while Y is gamma distributed. In the three other cases, the associated densities are provided. As a consequence, we obtain a new relation of convolution involving gamma distributions and Kummer distributions of type 2.