Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T01:58:26.251Z Has data issue: false hasContentIssue false
  • English
  • Français

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: martin.moehle@uni-tuebingen.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Angerer, W. P. (2001). An explicit representation of the Luria–Delbrück distribution. J. Math. Biol. 42, 145–174.CrossRefGoogle ScholarPubMed
Angerer, W. P. (2001). A note on the evaluation of fluctuation experiments. Mutation Res. 479, 207–224.CrossRefGoogle ScholarPubMed
Bartlett, M. S. (1978). An Introduction to Stochastic Processes, 3rd edn. Cambridge University Press.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (2000). Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego, CA.Google Scholar
Kemp, A. W. (1994). Comments on the Luria–Delbrück distribution. J. Appl. Prob. 31, 822–828.CrossRefGoogle Scholar
Kepler, T. B. and Oprea, M. (2001). Improved inference of mutation rates. I. An integral representation for the Luria–Delbrück distribution. Theoret. Pop. Biol. 59, 41–48.CrossRefGoogle ScholarPubMed
Kepler, T. B. and Oprea, M. (2001). Improved inference of mutation rates. II. Generalization of the Luria–Delbrück distribution. Theoret. Pop. Biol. 59, 49–59.CrossRefGoogle Scholar
Lea, D. E. and Coulson, C. A. (1949). The distribution of the number of mutants in bacterial populations. J. Genet. 49, 264–285.CrossRefGoogle ScholarPubMed
Lukacs, E. (1970). Characteristic Functions, 2nd edn. Griffin, London.Google Scholar
Luria, S. E. and Delbrück, M. (1943). Mutations of bacteria from virus sensitivity to virus resistance. Genetics 28, 491–511.CrossRefGoogle ScholarPubMed
Ma, W. T., Sandri, G. vH. and Sarkar, S. (1992). Analysis of the Luria–Delbrück distribution using discrete convolution powers. J. Appl. Prob. 29, 255–267.CrossRefGoogle Scholar
Pakes, A. G. (1993). Remarks on the Luria–Delbrück distribution. J. Appl. Prob. 30, 991–994.CrossRefGoogle Scholar
Prodinger, H. (1996). Asymptotics of the Luria–Delbrück distribution via singularity analysis. J. Appl. Prob. 33, 282–283.CrossRefGoogle Scholar
Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar