I was wondering if it is true that : $X$ and $Y$ gaussian and $\mathbb E[XY]=0\implies X$ and $Y$ independent ?
Is it true that $X$ and $Y$ gaussian and $\mathbb E[XY]=0\implies X$ and $Y$ independent?
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0No, you need that they are jointly gaussian. – 2017-01-04
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no, this is false.
If $X$ and $Y$ are jointly Gaussian though, the their covariance is $0$, then they are independent.
So you need to add the jointly Gaussian assumption and a the fact that one of them has $0$ mean (in this case $E(XY)=Cov(XY)$)
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0Thank you for your answer, but what mean *jointly Gaussian* ? – 2017-01-04
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0it means that the joint distribution of the vector $(X,Y)$ is a two dimensional Gaussian distribution – 2017-01-05