CORRELATIONS IN BINARY SEQUENCES AND A GENERALIZED ZIPF ANALYSIS

We investigate correlated binary sequences using an n-tuple Zipf analy sis, where we define ''words'' as strings of length n, and calculate t he normalized frequency of occurrence omega(R) of ''words'' as a funct ion of the word rank R. We analyse sequences with short-range Markovia n correlations, as well as those with long-range correlations generate d by three different methods: inverse Fourier transformation, Levy wal ks, and the expansion-modification system. We study the relation betwe en the exponent alpha characterizing long-range correlations and the e xponent zeta characterizing power-law behavior in the Zipf plot. We al so introduce a function P(omega), the frequency density, which is rela ted to the inverse Zipf function R(omega), and find a simple relations hip between zeta and psi, where omega(R) similar to R(-zeta) and P(ome ga) similar to omega(-psi). Further, for Markovian sequences, we deriv e an approximate form for P(omega). Finally, we study the effect of a coarse-graining ''renormalization'' on sequences with Markovian and wi th long-range correlations.