Equivalence of the Markov chains and two-sided symbolic sequences

A new object of the probability theory, a two-sided chain of events (symbols), is introduced. A theory of multi-steps Markov chains with long-range memory proposed in Phys. Rev. E, 68 (2003) 061107, is developed and used to establish the correspondence between these chains and two-sided sequences. The Markov chain is proved to be statistically equivalent to the definite two-sided one and vice versa. The results obtained for the binary chains are generalized to the chains of multi-valued symbols. Applicability of the two-sided chains to the examination of the one-dimensional Ising chains of classical spins is discussed.