CORRELATIONS IN BINARY SEQUENCES AND A GENERALIZED ZIPF ANALYSIS

We investigate correlated binary sequences using an n-tuple Zipf analysis, where we define ''words'' as strings of length n, and calculate the normalized frequency of occurrence omega(R) of ''words'' as a function of the word rank R. We analyse sequences with short-range Markovian correlations, as well as those with long-range correlations generated by three different methods: inverse Fourier transformation, Levy walks, and the expansion-modification system. We study the relation between the exponent alpha characterizing long-range correlations and the exponent zeta characterizing power-law behavior in the Zipf plot. We also introduce a function P(omega), the frequency density, which is related to the inverse Zipf function R(omega), and find a simple relationship between zeta and psi, where omega(R) similar to R(-zeta) and P(omega) similar to omega(-psi). Further, for Markovian sequences, we derive an approximate form for P(omega). Finally, we study the effect of a coarse-graining ''renormalization'' on sequences with Markovian and with long-range correlations.