  It is well-known that a Lorenz curve derived from the distribution function
of a random variable can be viewed itself as a probability distribution
function of a new random variable [3]. We prove the surprising result that a
sequence of consecutive iterations of this map leads to a non-corner case
convergence independent of the starting random variable. In the primal case,
the limiting distribution has a power law, with coefficient equal to the golden
section. In the reflected case, the limiting distribution is classical Pareto,
with a conjugate coefficient. Possible directions for applications are
discussed.
