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Convergence results for compound Poisson distributions and applications to the standard Luria–Delbrück distribution

Part of: Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  14 July 2016

M. Möhle
M. Möhle*
Affiliation:
Eberhard Karls Universität Tübingen
*
∗Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: [email protected]
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Abstract

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We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

Keywords

Compound Poisson distributioncontinuous Luria–Delbrück distributionFourier transformstable distributionweak convergence

MSC classification

Primary: 60F05: Central limit and other weak theorems 92D15: Problems related to evolution
Secondary: 60E10: Characteristic functions; other transforms 92D10: Genetics
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 620 - 631
Copyright
© Applied Probability Trust 2005 

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