  We give conditions under which a scalar random variable T can be coupled to a
random scaling factor $\xi$ such that T and $\xi$T are rendered stochastically
independent. A similar result is obtained for random measures. One consequence
is a generalization of a result by Pitman and Yor on the Poisson-Dirichlet
distribution to its negative parameter range. Another application are diffusion
excursions straddling an exponential random time.
