We prove the following theorem. Let $\alpha =a+ib$ be a nonzero complex number. Then the following statements hold: (i) Let either $b\ne 0$ or $b=0$ and $a>0$. Let ${\xi }_1$ and ${\xi }_2$ be independent complex random variables. Assume that the linear forms $L_1={\xi }_1+{\xi }_2$ and $L_2={\xi }_1+\alpha {\xi }_2$ are independent. Then ${\xi }_j$ are degenerate random variables. (ii) Let $b=0$ and $a<0$. Then there exist complex Gaussian random variables in the wide sense ${\xi }_1$ and ${\xi }_2$ such that they are not complex Gaussian random variables in the narrow sense, whereas the linear forms $L_1={\xi }_1+{\xi }_2$ and $L_2={\xi }_1+\alpha {\xi }_2$ are independent.

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关于复数和四元数随机变量的Skitovich-Darmois定理

我们证明以下定理。让$\alpha =\mathrm{一种}+\mathrm{一世}b$为非零复数。然后，以下语句成立：（i）任一个$b\ne 0$ 要么 $b=0$ 和 $\mathrm{一种}>0$。让${\xi }_{\mathrm{1个}}$ 和 ${\xi }_2$是独立的复杂随机变量。假设线性形式 ${\mathrm{大号}}_{\mathrm{1个}}={\xi }_{\mathrm{1个}}+{\xi }_2$ 和 ${\mathrm{大号}}_2={\xi }_{\mathrm{1个}}+\alpha {\xi }_2$是独立的。然后 ${\xi }_{Ĵ}$是退化的随机变量。（ii）让$b=0$ 和 $\mathrm{一种}<0$。然后存在广义的复杂高斯随机变量${\xi }_{\mathrm{1个}}$ 和 ${\xi }_2$ 这样它们在狭义上就不是复杂的高斯随机变量，而线性形式 ${\mathrm{大号}}_{\mathrm{1个}}={\xi }_{\mathrm{1个}}+{\xi }_2$ 和 ${\mathrm{大号}}_2={\xi }_{\mathrm{1个}}+\alpha {\xi }_2$ 是独立的。