A link between the Matsumoto-Yor property and an independence property on trees (Q2495416)

This paper deals with the so-called Matsumoto-Yor property. Let \(X\) and \(Y\) be two non-dirac, positive and independent random variables. A necessary and sufficient condition for the random variables \((X+Y)^{-1}\) and \(X^{-1}-(X+Y)^{-1}\) to be independent is that \(X\) follows a generalized inverse Gaussian distribution while \(Y\) is gamma-distributed with suitable parameters. The sufficiency part of this assertion is named in the related recent literature as Matsumoto-Yor property. The origin of the present paper is the observation of the form of the random variables \(U=(X+Y)^{-1}\) and \(V=X^{-1}-(X+Y)^{-1}\), which look like random variables dealt with in a previous paper by \textit{O. E. Barndorff-Nielsen} and the author [Adv. Appl. Probab. 30, No.~2, 409--424 (1998; Zbl 0912.60017)]. In this paper author explains how the result of Barndorff-Nielsen and himself implies a particular case of the Matsumoto-Yor property.