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Lumpability for non-irreducible finite markov chains

  • Atef M. Abdel-Moneim (a1) and Frederick W. Leysieffer (a2)

Abstract

Conditions under which a function of a finite, discrete-time Markov chain, X(t), is again Markov are given, when X(t) is not irreducible. These conditions are given in terms of an interrelationship between two partitions of the state space of X(t), the partition induced by the minimal essential classes of X(t) and the partition with respect to which lumping is to be considered.

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Postal address: West Heliopolis Post Office, 8 Ahmed Abdel Salam Zaki Street, Apartment #5, Cairo, Egypt.
∗∗ Postal address: Department of Statistics and Statistical Consulting Center, The Florida State University, Tallahassee, FL 32306, U.S.A.

References

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[1] Burke, C. J. and Rosenblatt, M. (1958) A Markovian function of a finite Markov chain. Ann. Math. Statist. 29, 11121122.
[2] Chung, Kai Lai (1967) Markov Chains, 2nd edn. Springer-Verlag, New York.
[3] Hachigian, J. and Rosenblatt, M. (1963) Collapsed Markov chains and the Chapman–Kolmogorov equation. Ann. Math. Statist. 34, 233237.
[4] Kemeny, J. G. and Snell, J. L. (1978) Finite Markov Chains, 2nd edn. Springer-Verlag, New York.
[5] Leysieffer, F. W. (1967) Functions of finite Markov chains. Ann. Math. Statist. 38, 206212.

Keywords

Lumpability for non-irreducible finite markov chains

  • Atef M. Abdel-Moneim (a1) and Frederick W. Leysieffer (a2)

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