We introduce a generalised multivariate Polya process for document language modelling. The framework outlined here generalises a number of statistical language models used in information retrieval for modelling document generation. In particular, we show that the choice of replacement matrix M ultimately defines the type of random process and therefore defines a particular type of document language model. We show that a particular variant of the general model is useful for modelling term-specific burstiness. Furthermore, via experimentation we show that this variant significantly improves retrieval effectiveness over a strong baseline on a number of small test collections.