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Convergence Results for Compound Poisson Distributions and Applications to the Standard Luria-Delbrück Distribution
Journal of Applied Probability
Vol. 42, No. 3 (Sep., 2005), pp. 620-631
Published by: Applied Probability Trust
Stable URL: http://www.jstor.org/stable/30040845
Page Count: 12
You can always find the topics here!Topics: Fourier transformations, Random variables, Infinity, Perceptron convergence procedure, Distribution functions, Mathematical integrals, Log integral function, Lead, Mathematical theorems, Mathematical independent variables
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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-Delbriick 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.
Journal of Applied Probability © 2005 Applied Probability Trust