Polar Decomposition of Scale-Homogeneous Measures with Application to Lévy Measures of Strictly Stable Laws
Polar Decomposition of Scale-Homogeneous Measures with Application to Lévy Measures of Strictly Stable Laws
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尺度齐次测度的极地分解及其在严格稳定规律的Lévy测度中的应用
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
I. Molchanov
中科院分区：
文献类型：
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作者：
S. Evans;I. Molchanov
某个空间上的标度是正真实的数群的可测作用。一个可测空间上的测度被称之为对某个非零真实的数α-齐次的，如果任何可测集合的质量被任何因子t > 0 t>0 t是集合的原始质量的倍数t^{-α} t-α。本文较一般地证明了给定可测空间上的α-齐次测度，在该空间与其子集与正真实的数的笛卡尔积之间存在可测双射（即，一个“极坐标系”），使得这个双射对α-齐次测度的推进是第一个分量上的概率测度的乘积（即在“角”分量上）和在正半直线上（即在“径向”分量上）的α齐次测度。这一结果适用于强度措施的泊松过程中出现的Lévy-Khinchin-Itô-like表示的无限可分随机元素。本文证明了：如果凸锥中的严格稳定随机元容许一个级数表示为Poisson过程的点之和，则它必然有一个LePage表示为i.i.d.由正半直线上独立单位强度泊松过程的连续点缩放的圆锥的随机元素，每个点的幂为$$-frac{1}{alpha }$$-1α。
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 $$alpha $$α-homogeneous for some nonzero real number $$alpha $$α if the mass of any measurable set scaled by any factor $$t > 0$$t>0 is the multiple $$t^{-alpha }$$t-α of the set’s original mass. It is shown rather generally that given an $$alpha $$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 $$alpha $$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 $$alpha $$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 $$-frac{1}{alpha }$$-1α.