On the Skitovich–Darmois theorem for complex and quaternion random variables
Keyword(s):
AbstractWe prove the following theorem. Let {\alpha=a+ib} be a nonzero complex number. Then the following statements hold: (i) Let either {b\neq 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. (ii) 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.
1980 ◽
Vol 10
(1)
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pp. 19-25
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2015 ◽
Vol 2015
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pp. 1-7
1980 ◽
Vol 32
(6)
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pp. 483-489
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