Memory functions and correlations in additive binary Markov chains

A theory of additive Markov chains with a long-range memory, proposed earlier in Usatenko et al ( 2003 Phys. Rev. E 68 061107), is developed and used to describe statistical properties of long-range correlated systems. The convenient characteristics of such systems, memory functions and their relation to the correlation properties of the systems are examined. Various methods for finding the memory function via the correlation function are proposed. The inverse problem ( calculation of the correlation function by means of the prescribed memory function) is also solved. This is demonstrated for the analytically solvable model of the system with a step-wise memory function.