Vocabulary Growth Fundamentals: Bernstein Functions and Hausdorff Sequences

2026-08-29

AI Summary

论文通过Bernstein函数和Hausdorff序列模型探讨词汇增长理论，解决了词汇类型预期数量建模问题，并分析了该理论在不同随机过程下的局限性。

Abstract

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.

Problem

• vocabulary growth
• Bernstein functions
• Hausdorff sequences

Innovation

• Bernstein functions
• Hausdorff sequences
• vocabulary growth
• hapax rate model
• renewal processes