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

Abstract

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 $b\ne 0$, 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.