yes!
...and if X not \in [-1,1], Y is randomly sampled from a Gaussian normal until it lands outside [-1,1].
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Then Y is actually Gaussian normal, and the scatter plot of X vs. Y looks like a Ø with a hard line segment in the center...
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...but the outer O is a Gaussian blur ring.
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So X and Y are perfectly correlated in the center, and uncorrelated outside it.
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A less extreme example would be a scatter plot where the points group close to the regression line in the middle but spread...
End of conversation
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for tail dependence, it needs to be the opposite. X=Y for extreme values, i.e. outside [-1,1].
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for tail independence, you don't need to do anything special to the joint gaussian. it already is tail independent.
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I was trying to describe asymptotic independence, as that is the term you brought into the discussion.
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And yes independence -> asymptotic independence but that's useless for describing what the latter is.
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Correlated Bivariate gaussians are tail independent.
End of conversation
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. Banned in Sweden. SubGenius, Zhuangist, white-hat troll. Defrocked mathematician. Brain problems.