On the Skitovich–Darmois theorem for complex and quaternion random variables

Abstract

Abstract We prove the following theorem. Let α = a + i ⁢ b {\alpha=a+ib} be a nonzero complex number. Then the following statements hold: (i) Let either b ≠ 0 {b\neq 0} or b = 0 {b=0} and a > 0 {a>0} . Let ξ 1 {\xi_{1}} and ξ 2 {\xi_{2}} be independent complex random variables. Assume that the linear forms L 1 = ξ 1 + ξ 2 {L_{1}=\xi_{1}+\xi_{2}} and L 2 = ξ 1 + α ⁢ ξ 2 {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent. Then ξ j {\xi_{j}} are degenerate random variables. (ii) Let b = 0 {b=0} and a < 0 {a<0} . Then there exist complex Gaussian random variables in the wide sense ξ 1 {\xi_{1}} and ξ 2 {\xi_{2}} such that they are not complex Gaussian random variables in the narrow sense, whereas the linear forms L 1 = ξ 1 + ξ 2 {L_{1}=\xi_{1}+\xi_{2}} and L 2 = ξ 1 + α ⁢ ξ 2 {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent.