  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
model allows exact analytical treatment. 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 transformation
of the chain parameters is presented. The diffusion 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.
