B1553
Title: Independence properties of the Kummer distribution and related characterizations
Authors: Jacek Wesolowski - Central Statistical Office (Poland) [presenting]
Abstract: A transformation of a pair of univariate random variables with Kummer distributions is presented, which preserves the independence property. This result covers, as limiting cases, several well-known independence-preserving properties, e.g., the Lukacs property, the Matsumoto-Yor property and several others. The main result is a characterization of the Kummer distribution based on this independence property. It is a generalization of an earlier characterization of the Kummer and Gamma laws based on independence properties. The proof of the characterization parallels one for the generalized Matsumoto-Yor property. However, in the case of the generalized Matsumoto-Yor property, a transformation preserving independence in the matrix-variate case was identified. In the Kummer case, identifying such transformation in the matrix-variate case remains an open problem. It is worth emphasising that the Kummer independence preserving transformation we present is equivalent to one of three types of the so-called quadrirational Yang-Baxter maps related to the preservation of independence property, which have been recently identified.
