Discrete random probability measures are a key ingredient of Bayesian nonparametric inference. A sample generates ties with positive probability and a fundamental object of both theoretical and applied interest is the corresponding 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 distribution for the success probability on the asymptotic behavior of the number of distinct values. A whole range of logarithmic behaviors are obtained by appropriately tuning the prior. A two-term expansion is also derived and illustrated 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.

离散随机概率测度是贝叶斯非参数推断的关键要素。样本产生具有正概率的联系，理论上和应用上都感兴趣的基本对象是不同值的相应数量。可以从小频率的衰减率确定增长率，这意味着，当递减的频率允许以易处理的形式出现时，可以方便地评估不同值的渐近性。我们专注于破坏几何的过程，我们研究了成功概率的分布对不同值数量的渐近行为的影响。通过对先验进行适当的调整，可以获得对数行为的整个范围。