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 ą 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´1 α .

Non-symmetric stable operators: regularity theory and integration by parts

“…We stress that γ and γ * in (1. 19) and (1.20) are functions defined on ∂Ω, not constants. Notice also that, since Ω is C 1,α and the Fourier symbol of L is Hölder continuous, then γ(z) and γ * (z) are Hölder continuous.…”

“…which means that µ is α-homogeneous (the interested reader may also look at Lemma 3.10 in [19] for a general approach).…”

Tail Measures and Regular Variation

“…Following [1] (which is motivated by the findings of [19,30]), where a boundedness along with the chosen group action plays a crucial role, we discuss first RV of Borel measures on general Polish metric spaces. In the light of F2, the investigation of −α-homogeneous measures, which are recently studied in an abstract framework in [31], is indispensable.…”

“…Homogeneous measures are studied under an abstract framework in [31] and as shown in [2,3,10,13,23,26] both Borel and cylindrical −α-homogeneous measures play a crucial role in the asymptotic theory of regularly varying random processes. In this section we shall assume A1) and continue with the definition and basic properties of tail measures.…”

“…In view of [31][Thm 1] we can find a D-valued random element Z such that ν = ν Z , provided that M0)-M2) hold; we give a different proof in Lemma 3.12.…”