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Independent normal distributions
If two random variables Z1 and Z2 have a joint normal distribution and have covariance zero, then how do you show that they're independent?I would guess it's quite easy, and ordinarily I'd just go look it up, but I don't have access to the library yet and google doesn't show up much (aside from a wikipedia article stating the fact but giving no proof).Cheers.
Independent normal distributions
No, the OP is correct, for normal multivariate distributions the converse *also* holds. (Sorry, no proof)
Independent normal distributions
No problem, as it happens I found the proof in a book on my desk (Vol II of Shreve's finance text).It follows from the fact that the multivariate density function factorises into seperate density functions when the correlation is zero.