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 $α=a+ib$ be a nonzero complex number. Then the following statements hold. $1$. Let either $b\ne 0$, or $b=0$ and $a>0$. Let $ξ_1$ and $ξ_2$ be independent complex random variables. Assume that the linear forms $L_1=ξ_1+ξ_2$ and $L_2=ξ_1+αξ_2$ are independent. Then $ξ_j$ are degenerate random variables. $2$. Let $b=0$ and $a<0$. Then there exist complex Gaussian random variables in the wide sense $ξ_1$ and $ξ_2$ such that they are not complex Gaussian random variables in the narrow sense, whereas the linear forms $L_1=ξ_1+ξ_2$ and $L_2=ξ_1+αξ_2$ are independent. We also study an analogue of the Heyde theorem for complex random variables.