Binary $N$-Step Markov Chains and Long-Range Correlated Systems

A theory of systems with long-range correlations based on the consideration of binary $N$-step Markov chains is developed. In our model, the conditional probability that the $i$th symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding $N$ symbols. The correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length $L$ are obtained analytically and numerically. If the persistent correlations are not extremely strong, the variance is shown to be nonlinearly dependent on $L$. A self-similarity of the studied stochastic process is revealed. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.

Figure 1

The numerical simulation of the dependence $D(L)$ for the generated sequence with $N=100$ and $\mu =0.4$ (circles). The correspondent value of the parameter $\gamma$ is 2.32. The solid line is the plot of function Eq. (7) with the same values of $N$ and $\mu$. The thin solid line describes the noncorrelated Brownian diffusion, $D(L)=L/4$. Squares and circles in the inset correspond to the stochastic and deterministic reduction, respectively (the parameter of decimation $\lambda =0.5$).

Figure 2

The $D(L)$ dependence for the coarse-grained texts of a collection of works on computer science ($m=2.2\times {10}^{-3}$, solid line), the Bible in Russian ($m=1.9\times {10}^{-3}$, dashed line), the Bible in English ($m=1.5\times {10}^{-3}$, dotted line), History of Russians in the 20th Century, by Oleg Platonov ($m=6.4\times {10}^{-4}$, dash-dotted line), and Alice’s Adventures in Wonderland, by Lewis Carroll ($m=2.7\times {10}^{-4}$, dash-dot-dotted line). In the upper inset: the $D(L)$ dependence for the coarse-grained text of the Bible (solid line) and for decimated sequences with different parameters $\lambda$: $\lambda =3/4$ (squares), $\lambda =1/2$ (stars), and $\lambda =1/256$ (triangles). In the lower inset: the antipersistent portion of the $D(L)$ plot for the Bible is magnified.

Figure 3

The $D(L)$ dependence for the coarse-grained DNA text of Bacillus subtilis, complete genome for three nonequivalent kinds of the mapping. Solid, dashed, and dash-dotted lines correspond to the mappings $\{A,G\}\to 0$, $\{C,T\}\to 1$ (the parameter $m=4.1\times {10}^{-2}$), $\{A,T\}\to 0$, $\{C,G\}\to 1$ ($m=2.5\times {10}^{-2}$), and $\{A,C\}\to 0$, $\{G,T\}\to 1$ ($m=1.5\times {10}^{-2}$), respectively.