Memory function versus binary correlator in additive Markov chains

We study properties of the additive binary Markov chain with short and long-range correlations. A new approach is suggested that allows one to express global statistical properties of a binary chain in terms of the so-called memory function. The latter is directly connected with the pair correlator of a chain via the integral equation that is analyzed in great detail. To elucidate the relation between the memory function and pair correlator, some specific cases were considered that may have important applications in different fields.