Equivalence of Markov's Symbolic Sequences to Two-Sided Chains
  A new object of the probability theory, two-sided chain of events (symbols), is introduced. A theory of multi-steps Markov chains with long-range memory, proposed earlier in Phys. Rev. E 68, 06117 (2003), is developed and used to establish the correspondence between these chains and two-sided ones. 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 taking on the arbitrary number of states. 
S. S. Apostolov, Z. A. Mayzelis, O. V. Usatenko, V. A. Yampol'skii