Equivalence of the Markov chains and two-sided symbolic sequencesS. S. Apostolov1, Z. A. Mayzelis1, O. V. Usatenko2 and V. A. Yampol'skii2
1 V. N. Karazin Kharkov National University - 4 Svoboda Sq., Kharkov 61077, Ukraine
2 A. Ya. Usikov Institute for Radiophysics and Electronics Ukrainian Academy of Science - 12 Proskura Street 61085 Kharkov, Ukraine
received 9 June 2006; accepted in final form 31 October 2006
published online 29 November 2006
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.
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.
02.50.Ga - Markov processes.
87.10.+e - General theory and mathematical aspects.
© EDP Sciences 2006