Additive N-step Markov chains as prototype model of symbolic stochastic dynamical systems with long-range correlations

Z.A. Mayzelis, S.S. Apostolov, S.S. Melnyk, O.V. Usatenko and Yampol’skii, V.A.

Chaos, Solitons & Fractals, 2007, vol. 34, issue 1, 112-128

Abstract: A theory of symbolic dynamic systems with long-range correlations based on the consideration of the binary N-step Markov chains developed earlier in Phys Rev Lett 2003;90:110601 is generalized to the biased case (non-equal numbers of zeros and unities in the chain). In the model, the conditional probability that the ith symbol in the chain equals zero (or unity) is a linear function of the number of unities (zeros) 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 verified by numerical simulations. A self-similarity of the studied stochastic process is revealed and the similarity group transformation of the chain parameters is presented. The diffusion Fokker–Planck equation governing the distribution function of the L-words is explored. If the persistent correlations are not extremely strong, the distribution function is shown to be the Gaussian with the variance being nonlinearly dependent on L. An equation connecting the memory and correlation function of the additive Markov chain is presented. This equation allows reconstructing a memory function using a correlation function of the system. Effectiveness and robustness of the proposed method is demonstrated by simple model examples. Memory functions of concrete coarse-grained literary texts are found and their universal power-law behavior at long distances is revealed.

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:34:y:2007:i:1:p:112-128

DOI: 10.1016/j.chaos.2007.01.054

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