一个Marchenko-Pastur定律的常数回归特征
A constant regression characterization of a Marchenko-Pastur law
摘要：
研究了自由概率下Marchenko-Pastur分布的Lukacs型刻画.证明了对于自由的$\mathbb{X}$和$\mathbb{Y}$，当给定$\mathbb{X+Y}$的$(\mathbb{X+Y})^{-1/2}\mathbb{X}(\mathbb{X+Y})^{-1/2}$的$1$和$-1$阶条件矩为常数时，则$\mathbb{X}$和$\mathbb{Y}$服从Marchenko-Pastur分布.
英文摘要：
Lukacs type characterization of Marchenko--Pastur distribution in free probability is studied here. We prove that for free $\mathbb{X}$ and $\mathbb{Y}$ when conditional moments of order $1$ and $-1$ of $(\mathbb{X+Y})^{-1/2}\mathbb{X}(\mathbb{X+Y})^{-1/2}$ given $\mathbb{X+Y}$ are constant then $\mathbb{X}$ and $\mathbb{Y}$ have Marchenko--Pastur distribution.
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