尺度齐性测度的极式分解及其在严格稳定律Lévy测度中的应用
Polar decomposition of scale-homogeneous measures with application to Lévy measures of strictly stable laws
摘要：
在某些空间上的标度是正实数群的可测作用。一个可测空间上的测度被称之为$α$ -齐次的，如果任何可测集合的质量被任何因子$t > 0$缩放，是集合原始质量的$t^{-α}$倍。给出了可测空间上的一个$α$ -齐次测度，在该空间与其子集和正实数的笛卡儿积之间存在一个可测双射（即，“极坐标系“），使得通过该双射的$α$齐次测量的前推是第一分量上的概率测度的乘积（即，在“角”分量上）和正半线（即，在“径向”分量上）上的$α$齐次测量。这个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。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.由正半线上的独立单位强度泊松过程的连续点缩放的圆锥的随机元，每个点的幂为$-\frac{1}α$。
英文摘要：
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 $-\frac{1}α$.
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