关于Matsumoto-Yor性质的推广
About an extension of the Matsumoto-Yor property
摘要：
如果$α,β>0$是不同的，如果$A$和$B$是独立的非退化的正随机变量，使$$S=\tfrac{1}{B}\,\tfrac{βA+B}{αA+B}\quad \mbox{and}\quad T=\tfrac{1}{A}\,\tfrac{βA+B}{αA+B} $$是独立的，我们证明了这种情况发生的当且仅当$A$和$B$有广义逆高斯分布与适当的参数。基本上，这已经在Bao和Noack（2021）中得到了证明，并补充了关于光滑密度存在的假设。这些问题的来源是Matsumoto和Yor（2001）对指数布朗运动的独立性的观察以及Croydon和Sasada（2000）对基于离散Korteweg-de Vries方程的随机递归模型的最近工作，其中上述结果被证明。我们还将直接结果扩展到随机矩阵，证明了独立的矩阵变量GIG变量满足上述独立性的矩阵变量模拟。GIG随机矩阵的特征通过这个独立性的问题仍然是开放的。
英文摘要：
If $α,β>0$ are distinct and if $A$ and $B$ are independent non-degenerate positive random variables such that $$S=\tfrac{1}{B}\,\tfrac{βA+B}{αA+B}\quad \mbox{and}\quad T=\tfrac{1}{A}\,\tfrac{βA+B}{αA+B} $$ are independent, we prove that this happens if and only if the $A$ and $B$ have generalized inverse Gaussian distributions with suitable parameters. Essentially, this has already been proved in Bao and Noack (2021) with supplementary hypothesis on existence of smooth densities. The sources of these questions are an observation about independence properties of the exponential Brownian motion due to Matsumoto and Yor (2001) and a recent work of Croydon and Sasada (2000) on random recursion models rooted in the discrete Korteweg - de Vries equation, where the above result was conjectured. We also extend the direct result to random matrices proving that a matrix variate analogue of the above independence property is satisfied by independent matrix-variate GIG variables. The question of characterization of GIG random matrices through this independence property remains open.
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