Operator-stable laws

Hudson, William N.; Mason, J. David

doi:10.1016/0047-259x(81)90086-5

Cited by 35 publications

(19 citation statements)

References 3 publications

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“…In the setting of Example 4.1, define the scaling by letting tx :" exptplog tqAux, t ą 0, for a non-degenerate matrix A. The corresponding stable elements are usually called operator stable, see [12,28]. An operator stable random element is infinitely divisible and so admits a series representation and a Lévy measure.…”

Section: Examplesmentioning

confidence: 99%

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

Evans

Molchanov

2017

J Theor Probab

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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 .

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Section: Examplesmentioning

confidence: 99%

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

Evans

Molchanov

2017

J Theor Probab

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“…In particular he showed that any operator stable measure + can be written as + G V + P where + G is a Gaussian measure and + P is an operator stable measure with no Gaussian component. Hudson and Mason (1981b) determined the support of each of these measures as follows. Select any exponent B of + and factor the minimal polynomial of B as g h where each root of g is simple and has real part equal to 1Â2 while each root of h has real part greater than 1Â2.…”

Section: Proofsmentioning

confidence: 99%