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The final σ-algebra of an inhomogeneous Markov chain with a finite number of states

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Abstract

It is proved that the final σ-algebra in the case of an inhomogeneous Markov chain with a finite number of states n is generated by a finite number (≤ n) of atoms. The atoms are characterized from the point of view of the behavior of trajectories of the chain. Sufficient conditions are given (in the case of a countable number of states) that there should exist an unique atom at infinity.

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Literature cited

  1. 1.

    V. Feller, Introduction to the Theory of Probability and Its Applications [in Russian], Vol. 1, Moscow (1967).

  2. 2.

    M. Loév, The Theory of Probability [in Russian], Moscow (1962).

  3. 3.

    D. V. Senchenko,“The characteristics of inhomogeneous Markov chains of processes with a finite number of states,” Teoriya Veroyatnostei i ee Primen.,13, 548–555 (1968).

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Translated from Matematicheskie Zametki, Vol. 12, No. 3, pp. 295–302, September, 1972.

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Senchenko, D.V. The final σ-algebra of an inhomogeneous Markov chain with a finite number of states. Mathematical Notes of the Academy of Sciences of the USSR 12, 610–613 (1972). https://doi.org/10.1007/BF01093996

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Keywords

  • Markov Chain
  • Finite Number
  • Countable Number
  • Unique Atom
  • Inhomogeneous Markov Chain