  We introduce an iterative discrete information production process where we
can extend ordered normalised vectors by new elements based on a simple affine
transformation, while preserving the predefined level of inequality, G, as
measured by the Gini index.
  Then, we derive the family of empirical Lorenz curves of the corresponding
vectors and prove that it is stochastically ordered with respect to both the
sample size and G which plays the role of the uncertainty parameter. We prove
that asymptotically, we obtain all, and only, Lorenz curves generated by a new,
intuitive parametrisation of the finite-mean Pickands' Generalised Pareto
Distribution (GPD) that unifies three other families, namely: the Pareto Type
II, exponential, and scaled beta distributions. The family is not only totally
ordered with respect to the parameter G, but also, thanks to our derivations,
has a nice underlying interpretation. Our result may thus shed a new light on
the genesis of this family of distributions.
  Our model fits bibliometric, informetric, socioeconomic, and environmental
data reasonably well. It is quite user-friendly for it only depends on the
sample size and its Gini index.
