  Discrete random probability measures are a key ingredient of Bayesian
nonparametric inferential procedures. A sample generates ties with positive
probability and a fundamental object of both theoretical and applied interest
is the corresponding random number of distinct values. The growth rate can be
determined from the rate of decay of the small frequencies implying that, when
the decreasingly ordered frequencies admit a tractable form, the asymptotics of
the number of distinct values can be conveniently assessed. We focus on the
geometric stick-breaking process and we investigate the effect of the choice of
the distribution for the success probability on the asymptotic behavior of the
number of distinct values. We show that a whole range of logarithmic behaviors
are obtained by appropriately tuning the prior. We also derive a two-term
expansion and illustrate its use in a comparison with a larger family of
discrete random probability measures having an additional parameter given by
the scale of the negative binomial distribution.
