Rank distributions of words in correlated symbolic systems and the Zipf law.

The binary many-step Markov chain with the step-like memory function is considered as a model for the analysis of rank distributions of words in correlated stochastic symbolic systems. We prove that this distribution obeys the power law with the exponent of the order of unity in the case of rather strong persistent correlations. The Zipf law is shown to be valid for the rank distribution of words with lengths about and shorter than the correlation length in the Markov sequence. A self-similarity in the rank distribution with respect to the decimation procedure is observed.