A growth model that generates an n-tuple Zipf law

The n-tuple Zipf law widely exists in real symbol sequences such as DNA and language. Based on the n-tuple Zipf analysis on a vast amount of empirical data, we propose a model that can generate sequences with n-tuple Zipf features. In simple words, it is an iterative copy and paste process: repeatedly select a random subsequence from the current sequence and attach it to the end. Numerical results of our model show that the n-tuple Zipf law exists in our model generated data. We give detailed analytical derivations of our model, and get two estimation equations: the Zipf exponent and the minimal length of n-tuple for Zipf law to appear. Our model can also reproduce the symmetry breaking process of ATGC number differences in DNA data.