Dependence of exponents on text length versus finite-size scaling for word-frequency distributions

Some authors have recently argued that a finite-size scaling law for the text-length dependence of word-frequency distributions cannot be conceptually valid. Here we give solid quantitative evidence for the validity of such scaling law, both using careful statistical tests and analytical arguments based on the generalized central-limit theorem applied to the moments of the distribution (and obtaining a novel derivation of Heaps law as a by-product). We also find that the picture of word-frequency distributions with power-law exponents that decrease with text length [Yan and Minnhagen, Physica A 444, 828 (2016)] does not stand with rigorous statistical analysis. Instead, we show that the distributions are perfectly described by power-law tails with stable exponents, whose values are close to 2, in agreement with the classical Zipfs law. Some misconceptions about scaling are also clarified.