Polar decomposition of scale-homogeneous measures with application to Lévy measures of strictly 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 $α$-homogeneous for some nonzero real number $α$ if the mass of any measurable set scaled by any factor $t > 0$ is the multiple $t^{-α}$ of the set's original mass. It is shown rather generally that given an $α$-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 $α$-homogeneous measure by this bijection is the product of a probability measure on the first component (that is, on the "angular" component) and an $α$-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 $-\frac{1}α$.