The growth statistics of Zipfian ensembles: Beyond Heaps’ law

We consider an evolving ensemble assembled from a set of n different elements via a stochastic growth process in which independent and identically distributed copies of the elements arrive randomly in time, and their statistics are governed by Zipf’s law. The associated “Heaps process” is the stochastic process tracking the fraction of different element copies present in the evolving ensemble at any given time point. For example, the evolving ensemble is a text assembled from a stream of words, and the Heaps process keeps count of the number of different words in the evolving text. A detailed asymptotic statistical analysis of the Heaps process, in the limit n→∞, is conducted. This paper establishes a comprehensive “Heapsian analysis” of the growth statistics of Zipfian ensembles. The analysis presented far extends and generalizes Heaps’ law, which asserts that the number of different words in a text of length l follows a power law in the variable l.