Note on a characterization of exponential distributions (Q1096270)

Let U be uniformly distributed on (0,1) and let Y and \(Y'=^{d}Y\) be random vectors with nonnegative components, U, Y and \(Y'\) independent. It is shown that the relation \(Y=^{d}U(Y+Y')\) is satisfied if and only if the components of Y are multiples of a single exponentially distributed random variable.