二元N步马尔可夫链的符号随机动力系统
Symbolic stochastic dynamical systems viewed as binary N-step Markov chains
摘要：
基于对二元N步马尔可夫链的考虑，发展了长程相关系统的理论。在该模型中，链中第i个符号等于零（或单位）的条件概率是前N个符号中单位数的线性函数。通过解析和数值计算，得到了任意长度L的字的相关函数、分布函数以及符号数的方差。揭示了所研究的随机过程的自相似性，给出了链参数的相似群变换。讨论了控制L字分布函数的扩散Fokker-Planck方程。如果持续相关性不是非常强，则分布函数是高斯分布，方差与L呈非线性相关。所开发的理论的适用性粗粒度的书面和DNA文本进行了讨论。
英文摘要：
A theory of systems with long-range correlations based on the consideration of binary N-step Markov chains is developed. In the 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. 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. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.
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