Asymptotic behavior of the number of distinct values in a sample from\n the geometric stick-breaking process

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