论文标题
Skitovich--Darmois和Heyde定理，用于复杂和季节随机变量
The Skitovich--Darmois and Heyde theorems for complex and quaternion random variables
论文作者
论文摘要
我们证明了经典的Skitovich--Darmois定理的以下类似物，用于复杂的随机变量。令$α= a+ib $为非零复数数。然后以下语句保留。 $ 1 $。令$ b \ ne 0 $，或$ b = 0 $和$ a> 0 $。令$ξ_1$和$ξ_2$为独立的复杂随机变量。假设线性形式$ l_1 =ξ_1+ξ_2$和$ l_2 =ξ_1+αξ_2$是独立的。然后$ξ_j$是退化的随机变量。 $ 2 $。令$ b = 0 $和$ a <0 $。然后在广泛的意义上存在复杂的高斯随机变量$ξ_1$和$ξ_2$，因此它们在狭窄的意义上不是复杂的高斯随机变量，而线性形式$ l_1 =ξ_1+ξ_2$和$ L_1+ξ_2$和$ l_2 =ξ_1+αξ_2$是独立的。我们还研究了Heyde定理的类似物，以进行复杂的随机变量。
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.
