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ZIPF AND LERCH LIMIT OF BIRTH AND DEATH PROCESSES

Published online by Cambridge University Press:  21 December 2009

B. Klar ,
P. R. Parthasarathy  and
N. Henze
B. Klar
Affiliation:
Institut für Stochastik, Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany E-mail: [email protected]
P. R. Parthasarathy
Affiliation:
Institut für Stochastik, Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany E-mail: [email protected]
N. Henze
Affiliation:
Institut für Stochastik, Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany E-mail: [email protected]
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Abstract

Birth and death processes are useful in a wide range of disciplines from computer networks and telecommunications to chemical kinetics and epidemiology. Data from many different areas such as linguistics, music, or warfare fit Zipf's law surprisingly well. The Lerch distribution generalizes Zipf's law and is applicable in survival and dispersal processes. In this article we construct a birth and death process that converges to the Lerch distribution in the limit as time becomes large, and we investigate the speed of convergence. This is achieved by employing continued fractions. Numerical illustrations are presented through tables and graphs.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 1 , January 2010 , pp. 129 - 144
Copyright
Copyright © Cambridge University Press 2009

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