论文标题
大约延伸原产
About an extension of the Matsumoto-Yor property
论文作者
论文摘要
如果$α，β> 0 $是不同的，如果$ a $和$ b $是独立的非脱位正随机变量 $ s = \ tfrac {1} {b} \，\ tfrac {βa+b} {αa+b} \ quad \ quad \ mbox {and} \ quad t = \ tfrac {1} {1} {a} {a} {a} {a} $$ 是独立的，我们证明这发生在且仅当$ a $ a $ a和$ b $具有适当参数的逆向高斯分布中。从本质上讲，这已经在Bao和Noack（2021）中证明了这一点，并具有关于平滑密度存在的补充假设。 这些问题的来源是关于Matsumoto and Yor（2001）引起的指数布朗运动独立性的观察，以及Croydon和Sasada（2000）在植根于离散的Korteweg -de Vries方程中的随机递归模型上的最新作品，其中上述结果被推测为上述结果。 我们还将直接结果扩展到随机矩阵，证明了上述独立性属性的矩阵变化类似物可以通过独立的矩阵变量变量来满足。通过这种独立性属性的演出随机矩阵的表征问题仍然开放。
If $α,β>0$ are distinct and if $A$ and $B$ are independent non-degenerate positive random variables such that $$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.
