Polar decomposition of scale-homogeneous measures with application to L\'evy measures of strictly stable laws

TL;DR

This paper develops a polar decomposition for scale-homogeneous measures, demonstrating a bijection to a product space, and applies it to Lévy measures of strictly stable laws, revealing their LePage series representations.

Contribution

It introduces a general polar decomposition for scale-homogeneous measures and applies it to Lévy measures, establishing new series representations for strictly stable laws.

Findings

Existence of a measurable bijection to polar coordinates for homogeneous measures

Application to Lévy measures of stable laws

Derivation of LePage series representations for stable laws

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 L\'evy-Khinchin-It\^o-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 $-\frac{1}{\alpha}$.