发表机构
Institute of Computer Science, Polish Academy of Sciences（波兰科学院计算机科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文综述词汇增长理论，整合伯恩斯坦函数与豪斯多夫序列理论并关联近期单例率模型，证明逻辑斯蒂单例率模型为伯恩斯坦函数，还分析了该理论的局限性。
AI 中文摘要
我们对建立在随机过程框架下的词汇增长理论进行综述。特别地，我们通过伯恩斯坦函数和豪斯多夫序列对类型的期望数量进行建模。这两类数学对象分别由导数或差分的交替符号定义，可分别与连续时间泊松点过程和离散时间独立同分布（IID）过程相关联。在已有词汇增长研究的基础上，我们整合了更广泛的伯恩斯坦函数理论与豪斯多夫序列理论，并将其与近期提出的单例率（hapax rate）模型相连接。具体而言，我们证明逻辑斯蒂单例率模型具有非负谱，因此它可定义为一个伯恩斯坦函数，从而解决了此前提出的问题。我们还通过考虑其在平稳过程和威布尔更新过程下的推广，分析了词汇增长的伯恩斯坦-豪斯多夫理论的局限性。
英文摘要
We survey the theory of vocabulary growth founded in the setting of stochastic processes. In particular, we model the expected number of types through Bernstein functions and Hausdorff sequences. These classes of mathematical objects, defined by alternating signs of their derivatives or differences, can be related to continuous-time Poisson point processes and discrete-time IID processes, respectively. Building on previous accounts of the vocabulary growth, we integrate the broader theories of Bernstein functions and Hausdorff sequences and connect them with recently developed hapax rate models. In particular, we prove that the logistic hapax rate model has a non-negative spectrum and hence it defines a Bernstein function, thereby solving an earlier posed problem. We also analyze the limitations of the Bernstein--Hausdorff theory of the vocabulary growth by considering its generalizations under stationary and Weibull renewal processes.
Comments41 pages
