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Taylor's law, via ratios, for some distributions with infinite mean

Part of: Distribution theory Distribution theory - Probability Applications Limit theorems

Published online by Cambridge University Press:  15 September 2017

Mark Brown ,
Joel E. Cohen  and
Victor H. de la Peña
Mark Brown*
Affiliation:
Columbia University
Joel E. Cohen*
Affiliation:
The Rockefeller University and Columbia University
Victor H. de la Peña*
Affiliation:
Columbia University
*
* Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA.
** Postal address: Laboratory of Populations, The Rockefeller University, New York, NY 10065, USA. Email address: [email protected]
* Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA.
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Abstract

Taylor's law (TL) originated as an empirical pattern in ecology. In many sets of samples of population density, the variance of each sample was approximately proportional to a power of the mean of that sample. In a family of nonnegative random variables, TL asserts that the population variance is proportional to a power of the population mean. TL, sometimes called fluctuation scaling, holds widely in physics, ecology, finance, demography, epidemiology, and other sciences, and characterizes many classical probability distributions and stochastic processes such as branching processes and birth-and-death processes. We demonstrate analytically for the first time that a version of TL holds for a class of distributions with infinite mean. These distributions, a subset of stable laws, and the associated TL differ qualitatively from those of light-tailed distributions. Our results employ and contribute to the methodology of Albrecher and Teugels (2006) and Albrecher et al. (2010). This work opens a new domain of investigation for generalizations of TL.

Keywords

Taylor's lawstable distributionLaplace transformTweedie distributionheavy tailcumulant generating functionapproximate exponentialitylog linear regression

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory 62P12: Applications to environmental and related topics 62P35: Applications to physics
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 657 - 669
Copyright
Copyright © Applied Probability Trust 2017 

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