  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. 90, 110601 (2003) is generalized to the biased case (non equal
numbers of zeros and unities in the chain). 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 (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.
