The Skitovich--Darmois and Heyde theorems for complex and quaternion random variables

We prove the following analogue of the classical Skitovich–Darmois theorem for complex random variables. Let $\alpha =a+ib$ be a nonzero complex number. Then the following statements hold. $1$. Let either $b\mathrm{\ne }0$, 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. $2$. 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. We also study an analogue of the Heyde theorem for complex random variables.