A scaling law beyond Zipf's law and its relation to Heaps' law

Abstract: The dependence with text length of the statistical properties of word occurrences has long been considered a severe limitation quantitative linguistics. We propose a simple scaling form for the distribution of absolute word frequencies which uncovers the robustness of this distribution as text grows. In this way, the shape of the distribution is always the same and it is only a scale parameter which increases linearly with text length. By analyzing very long novels we show that this behavior holds both for raw, unlemmatized texts and for lemmatized texts. For the latter case, the word-frequency distribution is well fit by a double power law, maintaining the Zipf's exponent value $\gamma\simeq 2$ for large frequencies but yielding a smaller exponent in the low frequency regime. The growth of the distribution with text length allows us to estimate the size of the vocabulary at each step and to propose an alternative to Heaps' law, which turns out to be intimately connected to Zipf's law, thanks to the scaling behavior.