  This work focuses on the characterization of the central tendency of a sample
of compositional data. It provides new results about theoretical properties of
means and covariance functions for compositional data, with an axiomatic
perspective. Original results that shed new light on the geostatistical
modeling of compositional data are presented. As a first result, it is shown
that the weighted arithmetic mean is the only central tendency characteristic
satisfying a small set of axioms, namely continuity, reflexivity and marginal
stability. Moreover, this set of axioms also implies that the weights must be
identical for all parts of the composition. This result has deep consequences
on the spatial multivariate covariance modeling of compositional data. In a
geostatistical setting, it is shown as a second result that the proportional
model of covariance functions (i.e., the product of a covariance matrix and a
single correlation function) is the only model that provides identical kriging
weights for all components of the compositional data. As a consequence of these
two results, the proportional model of covariance function is the only
covariance model compatible with reflexivity and marginal stability.
