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Biometrika 1960 47(3-4):235-244; doi:10.1093/biomet/47.3-4.235
© 1960 by Biometrika Trust
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Hierarchical birth and death processes I. Theory

GUNNAR BLOM

University of Lund Sweden

A hierarchical birth and death process is defined by dividing the states of a birth and death process into substates according to oertain rules. Only processes with constant transition probabilities are considered. In § 1, some basic results concerning integer-valued Markov processes are reviewed. In § 2, a special hierarchical process is discussed, and in § 3 equilibrium probabilities of the states of this process are derived in terms of given transition probabilities. In § 4, brief attention is paid to another special case. In § 5, a more general hierarchical process is analysed.

The processes considered in §§ 2–5 are finite. In § 6, this restriction is removed, so that the process is allowed to adopt an infinite denumerable number of states. Sufficient conditions for this general process to enter into statistical equilibrium are given.

Some of the results presented in this paper may be obtained from the general theory of reversible Markov chains and processes. However, the paper will be self-contained to a large extent, and most of the results will be derived from first principles.


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