  Galton's rank order statistic is one of the oldest statistical tools for
two-sample comparisons. It is also a very natural index to measure departures
from stochastic dominance. Yet, its asymptotic behaviour has been investigated
only partially, under restrictive assumptions. This work provides a
comprehensive {study} of this behaviour, based on the analysis of the so-called
contact set (a modification of the set in which the quantile functions
coincide). We show that a.s. convergence to the population counterpart holds if
and only if {the} contact set has zero Lebesgue measure. When this set is
finite we show that the asymptotic behaviour is determined by the local
behaviour of a suitable reparameterization of the quantile functions in a
neighbourhood of the contact points. Regular crossings result in standard rates
and Gaussian limiting distributions, but higher order contacts (in the sense
introduced in this work) or contacts at the extremes of the supports may result
in different rates and non-Gaussian limits.
