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Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1303–1321 | Cite as

Polar Decomposition of Scale-Homogeneous Measures with Application to Lévy Measures of Strictly Stable Laws

  • Steven N. Evans
  • Ilya Molchanov
Article

Abstract

A scaling on some space is a measurable action of the group of positive real numbers. A measure on a measurable space equipped with a scaling is said to be \(\alpha \)-homogeneous for some nonzero real number \(\alpha \) if the mass of any measurable set scaled by any factor \(t > 0\) is the multiple \(t^{-\alpha }\) of the set’s original mass. It is shown rather generally that given an \(\alpha \)-homogeneous measure on a measurable space there is a measurable bijection between the space and the Cartesian product of a subset of the space and the positive real numbers (that is, a “system of polar coordinates”) such that the push-forward of the \(\alpha \)-homogeneous measure by this bijection is the product of a probability measure on the first component (that is, on the “angular” component) and an \(\alpha \)-homogeneous measure on the positive half line (that is, on the “radial” component). This result is applied to the intensity measures of Poisson processes that arise in Lévy-Khinchin-Itô-like representations of infinitely divisible random elements. It is established that if a strictly stable random element in a convex cone admits a series representation as the sum of points of a Poisson process, then it necessarily has a LePage representation as the sum of i.i.d. random elements of the cone scaled by the successive points of an independent unit-intensity Poisson process on the positive half line each raised to the power \(-\frac{1}{\alpha }\).

Keywords

Disintegration Infinite divisibility LePage representation 

Mathametics Subject Classification (2010)

28A50 28C10 60B15 60E07 

Notes

Acknowledgements

The paper was initiated while SE was visiting the University of Bern supported by the Swiss National Science Foundation.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Statistics #3860, 367 Evans HallUniversity of CaliforniaBerkeleyUSA
  2. 2.Institute of Mathematical Statistics and Actuarial ScienceUniversity of BernBernSwitzerland

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